recursal/Fanatic-Fandom huggingface dataset:
Diego

> "*I'm on my way. Can't wait to see you!*"
> ― Diego on the phone
>
>

**Diego** is a civilian character who was introduced in S02E02.

## ***Biography***

### **S02E02**

Diego leaves his house in Los Injurus and is excited to meet with someone. Later on, many drivers along with Diego stop at a red light down a hill. Larry's boat plows into the set of cars stopped at the light. Diego is rendered immobile by a car pushed under him. A car catches fire which spreads to the other immobilized cars. Diego is still in his car trying to free it from being stuck. Charles bumps into him to free him and yells to abandon his car and save himself when the bump does nothing. Diego abandoned his car just as the flames reached it.
Zane

> "*Grabbed food, omw*"
> ― Zane texting
>
>

**Zane** is a civilian character who was introduced in the Season 2 Finale.

## ***Biography***

### **Season 2 Finale**

Zane leaves the diner in Belasco City after ordering food to go. Later on, an accident occurs where a runaway steam roller plows into a set of cars including Zane and crashes into a semi truck hauling a tanker trailer. Moments later, the tanker trailer explodes, knocking back most of the cars that are in its area and damaging several cars in the back. Zane survived as he escaped through the window of his car.
Warren

> "*Oh God, we're gonna be late again...*"
> ― Warren
>
>

**Warren** is a civilian character who was introduced in S02E08.

## ***Biography***

### **S02E08**

Warren and his wife turn left to enter the freeway. Later on the freeway, an accident occurs causing a pileup. Warren crashes into the cars in the pileup. Then, a Gavril T75 cement mixer crashes into the passenger side of Warren's car. Warren survived but his wife died.
Elizabeth

> "*Say another word and you'll be sleeping on the porch...*"
> ― Elizabeth
>
>

**Elizabeth** is a civilian character who was introduced in S02E02.

## ***Biography***

### **S02E02**

Elizabeth and Andrew depart from their house in Roane County, Tennessee. Andrew complains about spending a vacation visiting Elizabeth's parents and Elizabeth replied that he'll be sleeping on the porch if he says another word. Later on at the freeway, an accident occurs. Jefferson's car flips and rolls over several oncoming cars. Elizabeth crashes into Jefferson's car and flips over. She suffered serious injuries but lived.
The Hummer Driver

**The Hummer Driver** is a driver who drives a grey AM General Hummer. He was introduced in S02E08.

## ***Biography***

### **S02E08**

The Hummer driver drives through Lake Farsoe. Later at the construction site, the crane collapses due to overexertion and heavy winds and falls onto the freeway. The Hummer driver crashed into a BMW 740d and comes to rest near a tree. He survived with minor injuries.
The Kashira Driver

**The Kashira Driver** is a driver who drives a red Ibishu Kashira. He was introduced in Season 1 Finale.

## ***Biography***

### **Season 1 Finale**

The Kashira Driver leaves his house in Riverside. Joshua dropped a pallet of wood on the highway, causing an accident. The Kashira driver crashes directly into the pallet and rolled to a stop. He was injured but survived.
The Mustang Driver

**The Mustang Driver** was a driver who drove a black and red 1978 Ford Mustang King Cobra. She was introduced in the Season 1 Finale.

## ***Biography***

The Mustang Driver drives along American Road. Bernard ramps over a red Chevrolet Camaro and the gas tanks he was transporting exploded, launching onto the freeway below, where a few cars crash into it. The Mustang driver is unable to stop in time and shrieks as she rear ends a black Ibishu Blackfoot, followed by a bus smashing into her car. She didn't make it.
Alanna

> "*What is he doing?*"
> ― Alanna
>
>

**Alanna** is a civilian character who was introduced in the Season 2 Finale.

## ***Biography***

### **Season 2 Finale**

Alanna drives on the freeway in Las Colinias De Vapor. Angus drops his sandwich and accidently presses the button to raise the bucket trying to get it. Alanna notices the bucket raising and questions what's happening in front of her. Then several concrete blocks scatter onto the road as Alanna and the other drivers behind Angus drove over the blocks, causing them to lose control and crash into each other. Alanna was ok after the incident.
Alan

> "*Your junker is loaded and on the way ha ha :)*"
> ― Alan on the phone
>
>

**Alan** is a civilian character who was introduced in S01E04.

## ***Biography***

### **S01E04**

Alan drives past a blue Jeep Renegade struggling to climb a hill in the Polar Expressway. Later, he loses control on an icy hill road and crashes into two cars. Bill and two other driver stop by the crash to check if everyone is ok until Samantha slides down the road and crashes into Bill. A prison bus and sheriff crash into the pileup followed by Noah. Alan survived and was lucky to be alive.

### **S02E06**

Eight years prior to the Polar Expressway pileup, Alan hauls a trailer carrying a white 1974 Satsuma AMP through Baja Hills. He was calling someone and jokes that the caller's Junker is loaded and on the way. Later on, Richie and Marcus were arguing until Richie leaps over a hill and crashes into a BMW 525i and knocks the AMP off of Alan's trailer. This caused a huge pile up. Alan was ok.
Phil

> "*That didn't sound good.*"
> ― Phil
>
>

**Phil** is a truck driver who was introduced in S01E02.

## ***Biography***

### **S01E02**

Phil was hauls some logs in American Road. He decides to take a shortcut down a dirt road but regrets it due to the awful road conditions. When he go onto the road, he heard something smash, and said the noise didn't sound good. When Phil finally reaches the main road, he said he will never take the shortcut again. A set of cars were driving on the oncoming side and as Phil reached the set of cars, the connector on the trailer snapped. The trailer goes into oncoming traffic and smashes into the set of cars.
Passenger

> "*I can't wait to get off this bus. My legs are numb for sitting for so long.*"
> ― Passenger
>
>

The **Passenger** is a civilian character who was introduced in S02E06.

## ***Biography***

### **SO2E06**

The passenger rides on Jorge's bus in Belasco City. He expresses relief that they're almost at their destination, since their legs have gone numb. Later on, Doug rear ends Malcolm and is blown up by the explosive barrels Malcolm was carrying. The burning dump trucks rolls down the hill and plows into several cars, eventually crashing into the bus and knocking it over. It is unknown if the passenger survived.
Ray

> "*Yeah yeah, I know... Look, I gotta go. I'm driving right now. Talk to you later.*"
> ― Ray on the phone
>
>

**Ray** is a civilian character who was introduced in S02E03.

## ***Biography***

### **S02E03**

Ray drives in the tunnel in Utah. He tells someone on the phone that he will call them later because he is driving. Russell enters the freeway as Ray is exiting the tunnel. Russell merges into the inner lane not knowing Ray was there and Ray swerved. Ray was angry and brake checks Russell. Russell rear ends Ray and Ray rolls over into oncoming traffic. Tiffany and a yellow Ibishu Hopper crash into Ray's car. Ray survived with a few broken bones. His road rage killed two innocent people: the yellow Hopper driver and an elderly man driving a red ETK W-Series.
Eric

> "*No no no! Don't you do this to me! Not now! Not here!*"
> ― Eric
>
>

**Eric** is a civilian character who was introduced in S01E01.

## ***Biography***

### **S01E01**

Eric pulls a car carrier trailer in Beam Mountains. His van breaks down on a blind curve. Ben stops to help until David bumps into Eric's trailer and dangled helplessly from a cliff. David's truck detaches from its trailer and crashes down the hill. Eric was in shock.
The Chrysler Driver

**The Chrysler Driver** is a driver who drives a black 2012 Chrysler 300. He was introduced in S02E08.

## ***Biography***

### **S02E08**

The Chrysler driver drives through Lake Farsoe. Later at the construction site, the crane collapses due to overexertion and heavy winds and falls onto the highway. The Chrysler driver goes under the crane and screeches to a halt. He avoided all the carnage.
Forrest

> "*What do you want for dinner?*"
> ― Forrest
>
>

**Forrest** is a civilian character who was introduced in S03E01.

## ***Biography***

### **S03E01**

Forrest and Miranda leave a mountain overlook in Belasco City, discussing about dinner. Later on the highway, Wilson goes flying off the racetrack and hits a bendy bus, causing a pileup. Forrest is struck by several cars and crushed by a green ETK 6000-Series. He survived.
John

> "*Hey Mike what's up?*"
> ― John
>
>

**John** is a civilian character who was introduced in S01E02.

## ***Biography***

### **S01E02**

John gets back onto the freeway in Roane County, Tennessee after refilling his gas tank. Later on, he pulls over to the side of the road because he got a phone call from Mike. A GMC Syclone was parking backwards until it pushes a shopping cart down the hill. The shopping cart made a red Buick GNX swerve and rear ends John. John suffered minor injuries.
The Maserati Driver

**The Maserati Driver** was a driver who drove a yellow 2015 Maserati Ghibli. He was introduced in S02E08.

## ***Biography***

### **S02E08**

The Maserati driver drives though a tunnel in the German Alps. Alma cuts off Kaspar, causing a pileup at the mouth of the tunnel. The Maserati driver is unable to stop and crashes into the pileup. Klaus' brakes fail and he plows into the pileup at a high speed, crushing the Maserati driver and his passenger to death.
Miguel

> "*Finished working. I'll be home in 20.*"
> ― Miguel texting
>
>

**Miguel** was a truck driver who was introduced in S02E01.

## ***Biography***

### **S02E01**

Miguel finished work as he detaches from his trailer and drives home in East Coast, USA. Later on at a forked intersection, three cars and three trucks including Miguel crash into each other. Miguel was badly injured. Later on, Officer Henry arrived at the crash and tried to help Miguel but first aid was not enough and Miguel died on the way to the hospital.
Dexter

> "*Mr. Sallieri sends his regards.*"
> ― Dexter
>
>

**Dexter** was a criminal character who was introduced in S01E06.

## ***Biography***

### **S01E06**

Dexter follows Chris out of the parking garage in Los Injurus. Eventually, he goes beside Chris and rams him off the road. Chris falls onto a road and causes a multi-car accident that kills him and two innocent people. Dexter calls Chris a traitor and leaves the scene.

### **S02E07**

Dexter was hiding in the trees in Firwood Countryside waiting for Oscar and Michael's boss. When the convoy passes by, Dexter blows up the limo, killing the driver and boss. He then flees the scene and is chased by Michael and Oscar. A few police officers join the chase later on. Dexter causes several accidents, a few of which disable his pursuers. Dexter was still being chased by Officer Josh until Officer Kyle comes out of a dirt road and joins the chase. He catches up with Dexter and rams him off the road. Dexter rolls over, tumbles off a cliff and crashes into the rocks. His car explodes and Dexter is killed.

## ***Trivia***

* Dexter is most likely a hitman for hire.
* His line before pushing Chris off the road is a reference to Mafia 1 and 2, where Vito Scaletta said this line before killing Tommy.
* Dexter in the third scenario of SFD S02E07 has similarities with Hellhound from the main movie series;
	+ Both of them drive Dodge Chargers.
	+ Both of them have impressive driving skills
	+ Both of them got chased by main police characters (Officer Kyle for Dexter and HellCop for Hellhound).
	+ Both of them crashed their Dodge Chargers.
	+ Both of them got chased in a tight forest area (Firwood Countryside for Dexter and Altitude for Hellhound).
* His wrecked car was seen at a junkyard in Taken.
Sampling

The generalized differentiation is sampling. Let's say that John has 3 chickens and 2 ducks and he wants to remove 2 birds.

C
⋅
C
.
C
.
D
.
D

2
(
C
D
)

=

C
.
C
.
C
.
D
.
D

2
(
C
)
C
.
D
2
(
D
)

{\displaystyle { C \cdot C.C.D.D\over 2(C+D) }
 = { C.C.C.D.D\over { 2(C) + C.D + 2(D) } } }

=

C
.
C
.
C
.
D
.
D

2
(
C
)

C
.
C
.
C
.
D
.
D

C
.
D

C
.
C
.
C
.
D
.
D

2
(
D
)

{\displaystyle
 = { C.C.C.D.D\over { 2(C)} } + {C.C.C.D.D\over {C.D}} + { C.C.C.D.D\over {2(D)}} }

=

(

3
2

)

.
C
.
D
.
D
3.2.
C
.
C
.
D

(

2
2

)

.
C
.
C
.
C

{\displaystyle
 = {3\choose 2} .C.D.D + 3.2.C.C.D + {2\choose 2} .C.C.C }

=
C
.
D
.
D
C
.
C
.
D
C
.
C
.
C

{\displaystyle = 3.C.D.D + 6.C.C.D + 1.C.C.C }

One may conclude, for example, that the probability for John to remove two birds of different kind is 6/10.

in the above I have noted the differentiation by a ratio line and a set of 3 chickens by 3(C).

Eventually, we have reached in our research the formula of binomial coefficient.

X
.
X
.
.
.
.
.
X
.
X

{
X
,
X
,
.
.
.
,
X
}

=

X
n

k
(
X
)

=

(

n
k

)

.

X

n
−
k

{\displaystyle { X.X.....X.X \over \{ X,X,...,X \} }={ X^n \over k(X) } = {n\choose k}.X^{n-k} }

k-lists with and without repetition

X
.
X
.
.
.
.
.
X
.
X

{
X
,
X
,
.
.
.
,
X
}

=

X
n

{
X
}

k

=

(

n
k

)

.

X

n
−
k

{\displaystyle { X.X.....X.X \over \{ X,X,...,X \} }={ X^n \over { \lbrace X \rbrace }^k } = {n\choose k}.X^{n-k} }

X
.
X
.
.
.
.
.
X
.
X

X
.
.
.
.
.
X

=

X
n

X
k

=

n
!

(
n
−
k
)
!

.

X

n
−
k

{\displaystyle { X.X.....X.X \over X.....X }={ X^n \over X^k } = {n!\over (n-k)!}.X^{n-k} }

of course,

X
n

X
n

=
n
!
.1

{\displaystyle { X^n \over X^n} = n!.1 }
Ophelia

> "*Oh god, I hope I didn't forget my passport! Please be inside the purse!*"
> ― Ophelia
>
>

**Ophelia** was a civilian character who was introduced in S02E07.

## ***Biography***

### **S02E07**

Ophelia pulls over to the shoulder of River Highway to make sure that her passport was inside her purse. Later on at a crash site, Otto successfully flips a semi truck back on its wheels. However the truck starts to move as the parking brake was not on and the truck rolls towards the freeway. The truck rams Ophelia into the freeway and the drivers on the freeway crash into her and the semi truck. Ophelia's injuries were too great. She didn't survive.
Karen

> "*We raised five children. If they can't take care of our dog, we failed as parents, ha ha.*"
> ― Karen
>
>

**Karen** was an elderly civilian character who was introduced in S02E03.

## ***Biography***

### **S02E03**

Karen was in the passenger seat of Steven's car as they left their house to visit the placed they first met. Steven asks Karen if she locked the doors of their house. Karen replied that she did and says that no one is going to rob their house because they don't have anything expensive. Steven was driving slowly which causes George to overtake them and yell for them get off the road. Steven asks what his problem is but Karen replied to not worry about it since people nowadays are constantly in a hurry. Karen is excited about going back to where they met and says she always wanted to go back. Steven replies that he is excited as well. They pass by a car on a flat bed tow truck (Which was revealed to be FrIzErIs' Charger). Karen says it looked like a racecar. Steven mentions that a race is being held nearby and the driver (FrIzErIs) won't make it. Eventually, Steven and Karen wind up on the aforementioned race track and the racers crash into them. The medics rushed Karen and Steven to the hospital but their injuries were too serious. They both died the same day at almost the same time.
Larry

* Larry
* Larry (Rhino)

> "*Can't believe you're still talking about it...*"
> ― Larry
>
>

**Larry** is a civilian character who was introduced in S02E02.

## ***Biography***

### **S02E02**

Larry drives in Los Injurus pulling a trailer with a large dark red speedboat. Ellen was arguing with Larry on his decision to buy the boat. Larry and Ellen were going to the lake. Many drivers were stopping at a traffic light down the hill. Larry drives down the hill at a high rate of speed. Ellen tells Larry to slow down or else he'll kill somebody. Larry replies that he knows what he's doing. However, he can't slow down as the boat is heavy. Larry jackknifes and the boat plows the cars stopped at the traffic light. The impact sets a car on fire. One of the cars that was on fire explodes. Several drivers that were near the flames drive to a safe distance. Larry drives away to a safe distance.

## Larry

### Gender

Male

> "*This was the worst motel I've ever been to...*"
> ― Larry
>
>

**Larry** was a civilian character who was introduced in the Season 2 Finale.

## ***Biography***

### **Season 2 Finale**

Larry departs from a motel in Journey: The Long Road. He says that it was the worst motel he's been to. Later on as drivers are passing under an abandoned skyscraper, several concrete blocks fall of the skyscraper and cause a pile up. Larry got involved in the pile up and was crushed to death between two semi trucks.
Wilson

> "*I lost grip!*"
> ― Wilson
>
>

**Wilson** is a civilian character who was introduced in S03E01.

## ***Biography***

### **S03E01**

Wilson participates in a race in Belasco City. Teo's tire bursts, causing him to crash into the other racers. Wilson skids and flies off the track and onto the highway, hitting a bendy bus. He was in critical condition but survived.
Tommaso

> "*I can't talk right now, I'm driving. I'll be home soon*"
> ― Tommaso on the phone
>
>

**Tommaso** was a civilian character who was introduced in S02E09.

## ***Biography***

### **S02E09**

Tommaso drives in Italy. He is on the phone to someone saying that he can't talk since he's driving and that he'll be home soon. Later on the freeway, Ezio crashes into a rock dropped by a red MAN TGS and three cars on top of his trailer fall off onto the road, causing a pileup. Tommaso crashes head on into a red BMW. He didn't survive.

## ***Trivia***

* The name "Tommaso" is Italian for "Thomas".
Fiona

> "*God, I hate this weather... Can't wait for spring.*"
> ― Fiona
>
>

**Fiona** is a civilian character who was introduced in S02E09.

## ***Biography***

### **S02E09**

Fiona departs from her cabin in Tellude. She says that she hates the snowy weather and can't wait for spring. Later on the freeway, Simon flips over and blocks the freeway, causing a massive pileup. Fiona is unable to stop in time and crushed a red customized Volkswagen Golf before getting her rear-end crushed by a white Gavril T75 Dump Truck. She survived.
Brendan

> "*Thanks for stopping by.*"
> ― Brendan
>
>

**Brendan** is a civilian character who was introduced in S02E04.

## ***Biography***

### **Seconds From Disaster S02E04**

Brendan gets a flat tire and stops on a dirt road in Utah. Antwon encounters Brendan and stops beside him. Brendan tells Antwon that his friend is coming to help him and thanks Antwon for staying for a bit. Antwon leaves and drives over a few rocks slowly, however he unknowingly knocks one of them off the cliff. The rock rolls down and hits two other rocks which slide off the cliff. The rocks land on a few cars which causes a massive pileup on the road below and claims ten lives.
Jonathan

> "*How many times to I have to drag these truckers out of this exact same spot...*"
> ― Johnathan
>
>

**Jonathan** is a dump truck driver who was introduced in S02E04.

## ***Biography***

### **S02E04**

Jonathan drives along the road in Bumpy Island. Later on, he encounters Mark stuck on a hill. He stops in front of Mark and offers him help, which Mark accepts. Jonathan began to pull Mark up the hill, however the chain snaps and Mark slides down the hill. The tanker trailer dangles off a cliff and eventually detaches. The tanker trailer crashes into Rory and causes an explosion that kill sixteen people.
Harold

> "*Well this is harder than I thought...*"
> ― Harold
>
>

**Harold** was a dump truck driver who was introduced in S01E09.

## ***Biography***

### **S01E09**

Harold drives in the Open Cut Mine. He stops in front of Spencer. Harold exits his truck and asks Spencer what happened. Spencer tells Harold that his truck has a flat tire but he can move it out of the way. Harold claims that he can fit through. Harold tries to fit past until the dump truck falls off the road. Harold crashes down into the bottom of the mine crushing several vehicles. Harold didn't survive the fall.
Beth

The Rat Catcher

**Beth,** nicknamed *The Rat Catcher,* was a character on Season 1 of Total Drama Murders: Escape the Night

## Quotes

* "You can't be serious!"
* "But I don't wanna chose!"
* "I can beat it to death with a rock!"
* "Oh-no!"

## Trivia

* Beth is one of the two guests (throughout the whole TDM: Escape the Night) that appeared in a Total Drama Murders season, that being Total Drama Murders. The other guest that appeared in a Total Drama Murders season is Topher, in the 2nd season.
* Besides the introduction, Beth is the first guest to use the confessional this season.
* She was also the third contestant to appear in as a contestant in a second season, following Sky and Elizabeth.
Officer Greyson

> "*It's rolling straight into the highway!*"
> ― Officer Greyson
>
>

**Officer Greyson** is a police officer who was introduced in S02E07.

## ***Biography***

### **S02E07**

Officer Greyson was at a crash site in River Highway. Otto successfully flips the semi truck back on its wheels. However the truck starts to move as the parking brake was not on. Officer Greyson chases after the truck rolling straight into the freeway. The truck rams Ophelia onto the freeway and the drivers on the freeway crash into her and the semi. Many drivers crash into each other. Drivers on the oncoming side lose control and crash into each other. Officer Greyson accidently clips Jace's car and Jace crashes into the cars. Officer Greyson was ok and helped the injured.
Santiago

> "*Oh my God, I can't get out!!!*"
> ― Santiago
>
>

**Santiago** was a civilian character who was introduced in S02E02.

## ***Biography***

### **SO2E02**

Santiago stops at a red light in Los Injurus. Larry is unable to stop due to the weight of the boat and it plows into the set of stopped cars and starts a fire. Santiago's car was disabled. He was unable to get out of his car and burned to death.
Russell

> "*Nice cars they got here.*"
> ― Russell
>
>

**Russell** is a truck driver who was introduced in S02E03.

## ***Biography***

### **S02E03**

Russell exits a gas station hauling a tanker trailer in Utah. He passes by Inertia's Service Center as a cyan steel Dodge Viper SRT-10 ACR Custom enters the service center. Russell says that they have nice cars. Later on, he enters the freeway. Russell merges into the inner lane not knowing Ray was there and he swerved. Ray was angry and he brake checks Russell. Russell crashes into Ray and his Hopper rolls over into oncoming traffic. This causes a huge pileup. Russell was in a state of shock over what just happened.
Albert

> "*Slow down you moron!*"
> ― Albert
>
>

**Albert** is a civilian character who was introduced in S01E08.

## ***Biography***

### **S01E08**

Albert drives on a mountain road in Altitude. He passes a speeding Dodge Magnum and yells at the driver to slow down. Later on, he passes by Samuel's broken down truck when a rock tumbling down the mountain collides into the logs, launching them off of the trailer and down the mountain. The logs collide into the vehicles on the road below. Albert was not hit by the rock and pulls over to check up on Samuel.
Officer Franklin

> "*Pull over now, or I will use...*"
> ― Officer Franklin
>
>

**Officer Franklin** is a police officer who was introduced in S02E07.

## ***Biography***

### **S02E07**

Officer Franklin spots Dexter and joins with Oscar and Michael to pursue him in Firwood Countryside. Dexter jumps over a high jump and lands on the other side. Officer Franklin jumps over but crash lands onto Martinez. Officer Franklin survived.
Selena

> "*Move that thing out of the way!*"
> ― Selena
>
>

**Selena** is a civilian character who was introduced in S03E01.

## ***Biography***

### **S03E01**

Selena drives in Los Injurus. She gets stuck behind a slow MAN TGS and yells for it to move. Later on, Cameron is thrown off a bridge by Aztec and Raven and lands on a BMW Z4, causing a pileup. Selena slides off the road attempting to avoid the crash but hits a tree. She was ok.
Jeremy

> "*It's 8PM. Where are all these cars coming from?!*"
> ― Jeremy
>
>

**Jeremy** is a civilian character who was introduced in S02E03.

## ***Biography***

### **S02E03**

Jeremy leaves his apartment in Belasco City as he worries he is going to be late. Later on, Terry was walking on the streets humming to music until he gets hit by a trash container rolling down the hill. Jeremy swerved to avoid Terry and goes off of a Civetta Bolide and crashes into a white bus, seriously wounding the bus driver. The bus continues driving forward as the bus drivers foot is on the pedal, sideswiping several car before crushing Sebastian. Jeremy survived with non-life threatening injuries. If he just went straight, he wouldn't have hit Terry. There was enough space to drive around him. This mistake will haunt him for the rest of his life.

### **S02E06**

Fifteen years after the accident in Belasco City, Jeremy was driving through Baja Hills. He then encounters a pile up and damages his car. More cars crash into the cars in the pileup. Jeremy realizes the oncoming drivers can't see due to the blind hill and races to the top. He honks at the other drivers and yells for them to stop. A few of them slow down but few of them still crash into the cars. Jeremy eventually blocks the road, ending the carnage and possibly saving a lot of lives.

### **S02E08**

In 2022, Jeremy is driving along the freeway in Lake Farsoe. He questions the freeway is so busy at 8 in the evening. Later on at the construction site, the crane collapses due to heavy winds and overexertion. Jeremy crashes into several cars and comes to rest under the end of the crane. He survives the incident.
Zack

> "*Excavators aren't even here yet?! Someone is getting fired...*"
> ― Zack
>
>

**Zack** is a civilian character who was introduced in S01E09.

## ***Biography***

### **S01E09**

Zack overtakes a Gavril T75 dump truck and drives to the bottom of the Open Cut Mine. He reaches the bottom of the mine and finds all the workers waiting. He gets mad that the excavators are late and mutters that someone is getting fired. Zack calls Spencer as he was hauling the excavators. Spencer tells Zack that his truck got a flat tire and he's waiting for the service guy. Zack gets annoyed because the service guy is with him at the bottom of the mine. Before Zack can tell the service guy where Spencer is, Harold comes crashing into the mine and crushes the service guy along with two trucks and a bulldozer. Zack was one of four survivors.
Tim

> "*Was that an earthquake?*"
> ― Tim
>
>

**Tim** was a civilian character who was introduced in S01E03.

## ***Biography***

### **S01E03**

Tim drives through Sadler Gorge. An earthquake occurs. Tim crashes into a red Oldsmobile Delta 88 while on a bridge. Then the aftershock hit and the bridge collapses. Tim was still in his car as the bridge collapsed and fell to his demise.
Will

> "*Finally I can go home...*"
> ― Will
>
>

**Will** was a civilian character who was introduced in S01E06

## ***Biography***

### **S01E06**

Will detaches his truck from his trailer after delivering his cargo in Los Injurus and begins heading home. Later on, Dexter goes beside Chris and rams him off the road. Chris loses control and falls onto a blue Gavril H45 Box Truck. Will and a bus crushes Chris' car and Will crushes an orange Ibishu Covet. He was ok.

### **S02E01**

One month later, Will leaves the gas station in Los Injurus after refilling his gas tank. Later on the freeway, Kevin abandons his dog behind on the side of the freeway. The dog whines and barks as it wanders onto the road. Will notices the dog and he swerved to avoid it until his car goes dangling off a cliff. An RV avoids the dog but Randy was behind the RV and he swerves, flips over and knocks Will's car off the cliff. Will's car crushes Colby's van and explodes. Will had zero chance of survival.
Autica

**Autica**, auch die **Goldene Stadt** genannt, ist die Hauptstadt des Vereintes Kaiserreiches. In ihr befindet sich der Goldene Palast, Hauptsitz der Germanitas Solis und der kaiserlichen Dynastie. Autica behauptet von sich, ebenso wie Ambadale, die größte Stadt der Welt zu sein. Während die Hafenstadt in ihren Slums wohl mehr Menschen beheimatet, sind diese in der Regel nicht offiziell verzeichnet. Aus diesem Grund hat Autica nominell tatsächlich die meisten Einwohner.
Chris

> "*I'm on my way. Should be there in 10 minutes*"
> ― Chris on the phone
>
>

**Chris** was a civilian character who was introduced in S01E06.

## ***Biography***

### **S01E06**

Chris exits a parking garage in Los Injurus and is followed by Dexter. He calls someone to tell them he'll be at his destination in 10 minutes while Dexter follows him at a distance. Dexter goes beside Chris and rams him off the road. He loses control and falls onto a blue Gavril H45 Box Truck. Will and a bus crush Chris. His crushed car rolls into a Ibishu Hopper taxi. Chris was killed in the accident.

## ***Trivia***

* Chris betrayed Mr. Sallieri in some way and Dexter was hired to execute him.
* Chris drove a car similar to Tom from S01E01
Rosa

> "*There's a plane coming!*"
> ― Rosa in Italian
>
>

**Rosa** was a character who was introduced in the Season 1 Finale.

## ***Biography***

### **Season 1 Finale**

Rosa rides with Luca as they flee the police. They erratically weave through traffic until a Hirochi Sunburst Polizia joins the chase. Rosa and Luca arrive at the airport where they ditch their car and get in a plane. Their plane gets to the runaway but there was a cargo plane was about to land. Rosa says that there's a cargo plane coming but Luca says that they will make it. Rosa and Luca take off but they crash into the cargo plane, killing them both instantly.
Emily

> "*Oh my god, you got your license!*"
> ― Emily
>
>

**Emily** is a civilian character who was introduced in S02E05.

## ***Biography***

### **S02E05**

Emily and her husband Evan arrive at a plane hangar in Los Injurus. Emily was surprised that Evan got his plane license. They get out of Evan's pickup truck and got into his plane. The plane makes it to the runaway. As Evan takes off, he notices something off as there was not enough power for his plane. The wind is too strong and Evan decides to make an emergency landing on the freeway, however the plane crashes into a white Mercedes-Benz E63 AMG. Many drivers try swerve out of the way but crash into each other. A few drivers swerve to the oncoming side and crash into the other drivers. Emily survived with minor injuries.
Ron

> "*People must be going home after work.*"
> ― Ron
>
>

**Ron** is a civilian character who was introduced in S02E02.

## ***Biography***

### **S02E02**

Ron and Charlie recklessly race their rental supercars along the road in Italy. Charlie claims that renting the cars were a great idea. Ron replied to not crash it as he couldn't afford it. Charlie overtakes Ron but he goes the wrong way. Charlie turns around and drives behind Ron. Charlie comments on how busy the rural Italian road they're on is. Ron replies that people were likely driving home from work. Ron and Charlie enter the freeway. Ron tells Charlie that he got to the freeway first and Charlie lost the race. Charlie replied that he will win next time. Ron and Charlie race off down the freeway, oblivious to the pileup caused by Charlie.
Mike

> "*So we could get there on time, Megan*"
> ― Mike
>
>

**Mike** was a civilian character who was introduced in S01E05.

## ***Biography***

### **S01E05**

Mike and Megan drive in the dark at East Coast, USA. Megan asked why they had to get up early and Mike replied that they have to be somewhere in time. Later on, Ben was driving his until a blue Honda Civic flashes his high beams at him. Ben crashes into Archie's car and clips Mike. Mike pulled over to the side of the road until a Gavril Roamer knocks Mike's car off the road. Both the Roamer and the ETK tumbled down the hill. The ETK splashes into the water. Mike was unable to free himself and drowned.
Hydraponsid

**Hydraponsid** entsteht bei der Transmutation von reinem Quecksilber mit Pontium. Es ist ein tief violettes Pulver, welches von den Mercurii für alchemische Zwecke verwendet wird.

Das hochgiftige Hydrargyrum ist extrem schwer zu beherrschen und kann bei falscher Anwendung zum Tode des Nutzers führen. Doch von einem fähigen Alchemisten geführt zeigt es ein Potential, das alle anderen Ponside in den Schatten stellt. So kann es dazu genutzt werden, Seelen aus der Geisterwelt zu beschwören und in der physischen Welt zu binden. Den fähigsten Alchemisten gelingt es sogar, Tote wieder zum Leben zu erwecken. Die Nutzung von Hydraponsid wird auch als **Nekroalchemie** bezeichnet.
Ellen

> "*Can't believe you bought this stupid boat...*"
> ― Ellen
>
>

**Ellen** is a civilian characters who was introduced in S02E02.

## ***Biography***

### **S02E02**

Ellen rides with Larry in Los Injurus. Ellen was arguing with Larry over his decision to buy the speedboat. Ellen and Larry are going to a lake. Larry drives down the hill at a high rate of speed. Ellen tells Larry to slow down or else he'll kill somebody. Larry replies that he knows what he's doing. However, he can't slow down as the boat was heavy. Ellen screams as the boat plows into a set of cars stopped at the traffic light. The impact causes a car to catch fire. One of the cars that was on fire explodes. All drivers with mobile vehicles drove to a safe distance. Larry races to a safe distance, allowing Ellen to survive.
Matt

> "*No no no! Don't you die on me!*"
> ― Matt
>
>

**Matt** was a civilian character who was introduced in S01E09.

## ***Biography***

### **S01E09**

Matt enters a tunnel in Altitude. A blue Gavril T75 was driving in the tunnel until its engine bursts into flames. The drivers behind the truck including Matt stop. Matt spots the burning truck and tries to get out of the tunnel but his car dies. A MAN TGS hauling a tanker trailer enters the tunnel and plows into the cars inside the tunnel. There was oil leaking from the truck which made it burst into flames. The fire spreads to the other cars. Matt was still in his car and either perished in the flames or succumbed to smoke inhalation.
Metalle

Als die **Sieben Metallen**, auch **Planetenmetalle** genannt, sind die Grundstoffe der Alchemie bekannt. In Verbindung mit Pontium lassen sich aus ihnen Ponside gewinnen, die zu unterschiedlichen Zwecken eingesetzt werden. Jedes Metall korreliert mit einem Himmelskörper und wird von einer Bruderschaft verehrt und eingesetzt.

 Eisen (Ferrum)
 Gold (Aurum)
 Kupfer (Cuprum)
 Quecksilber (Hydrargyrum)
 Zinn (Stannum)
Richard

> "*It all happened so fast. I already called the cops.*"
> ― Richard
>
>

**Richard** is a civilian character who was introduced in S01E05.

## ***Biography***

### **S01E05**

Richard drives in the dark at East Coast, USA. Later on, a dark green Gavril D15 Extended crashes into Archie's car and rolls over. Richard avoids getting hit by the D15 and pulls over to the side of the road where he calls 911 on the accident. Later on, FrIzErIs was driving down the road until Richard flashed his high beams at him. FrIzErIs stopped in front of the crash and thanked Richard. FrIzErIs tells Richard to keep the high beams on as they check the cars to see if they can help anyone.
Pilot Jack

> "*Jesus. What just happened?*"
> ― Pilot Jack after the accident
>
>

**Pilot Jack** is a helicopter pilot who was introduced in S02E08.

## ***Biography***

### **S02E08**

Pilot Jack along with Medic Laura get into a rescue helicopter to respond to Joel's 911 call in Firwood Countryside. They fly above the town and then along the coastline, where they spot Joel's pickup. Officer Stephen and three other emergency vehicles locate Pilot Jack's helicopter and arrive at the location. Pilot Jack mentions that airlifting Joel would be a better option to do since Joel injured his leg. However an accident occurs on the freeway and Pilot Jack was shocked at what just happened. The medics, firefighters and police officers nearby helped people get out of their wrecked cars.

## ***Trivia***

* He is the first named helicopter pilot in the series.
Bill

> "*Is everyone ok?*"
> ― Bill
>
>

**Bill** is a civilian character who was introduced in S01E04.

## ***Biography***

### **S01E04**

Bill stops at a crash in Polar Expressway to check if everyone is ok in the Polar Expressway. Samantha slides down the road and crashes into him. A prison bus and sheriff crash into the pileup followed by Noah. Bill managed to run away from the crash site in time.
Ben

* Ben
* Ben (Blackfoot)

> "*I think we should try to push it further before someone crashes into your trailer **(...)** Hold on, I have to answer this.*"
> ― Ben
>
>

**Ben** is a civilian character who was introduced in S01E01.

## ***Biography***

### **S01E01**

Ben drives Beam Mountains. He spots Eric's broken down van. Ben stops to help him but had to answer a phone call. Later on, David bumps into Eric's trailer and dangled helplessly from a cliff. Ben tells David not to move as he will help him but it was too late as David's truck detaches from the trailer and crashes down the hill. Ben reacts in sheer horror.

## Ben

### Gender

Male

> "*Turn off the high beams you idi...*"
> ― Ben
>
>

**Ben** is a civilian character who was introduced in S01E05.

## ***Biography***

### **S01E05**

Ben and his wife drive through East Coast, USA. Ben gets blinded by a Honda Civic driver who left his high beams on and rear ends Archie's car. He swerves left, clips Mike, and falls off the cliff, smashing into a tree. Ben survived but his wife died on the way to the hospital. The Civic driver who blinded Ben was never found.
Fibonacci

## **Compositions with 1 or 2 elements**, F = 1 + X·F + 2(X)·F

A080599 Bob Proctor

```
1, 1, 3, 12, 66, 450, 3690, 35280, 385560,...

```

a

n

=
n
.

a

n
−
1

n
(
n
−
1
)

2

a

n
−
2

{\displaystyle a\_{n}=n.a\_{n-1}+{{n(n-1)} \over {2}}a\_{n-2}}

the types are

N/A
1
1+1, 2
1+1+1, 1+2, 2+1
1+1+1+1, 1+1+2, 1+2+1, 2+1+1, 2+2

1, 1, 3, 7, 25, 81, 331, 1303, 5937, 26785,

a

n

=

{\displaystyle a\_{n}=}

a

n
−
1

2
(
n
−
1
)

a

n
−
2

{\displaystyle a\_{n-1}+2(n-1)a\_{n-2}}

**Ens(X+X2)**

e
x
p
(
t
x
t

x

2

)

{\displaystyle exp(tx+tx^{2})}

1, 1, 4, 18, 120, 960, 9360,

n
!
⋅

F

n
1

{\displaystyle n!\cdot F\_{n+1}}

a

n

=

{\displaystyle a\_{n}=}

n

a

n
−
1

n
(
n
−
1
)

a

n
−
2

{\displaystyle na\_{n-1}+n(n-1)a\_{n-2}}

 A000085
1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496,
Involutions

a

n

=

{\displaystyle a\_{n}=}

a

n
−
1

(
n
−
1
)

a

n
−
2

{\displaystyle a\_{n-1}+(n-1)a\_{n-2}}

**Ens(X+E2(X))**
Bessel numbers of the second kind.

e
x
p
(
t
x
t

x

2

2

)

{\displaystyle exp(tx+t{\frac {x^{2}}{2}})}

l
i
n
(
k
x
k

x

2

2

)

{\displaystyle lin(kx+k{\frac {x^{2}}{2}})}

1, 1, 3, 12, 66, 450, 3690, 35280,

a

n

=

{\displaystyle a\_{n}=}

n

a

n
−
1

n
(
n
−
1
)

2

a

n
−
2

{\displaystyle na\_{n-1}+{n(n-1) \over 2}a\_{n-2}}

 A008619
1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7,
partitions of n in k parts of size 1 or 2

a
n

=

a

n
−
1

a

n
−
2

{\displaystyle a\_n = a\_{n-1} + a\_{n-2} }

## the native construction H = 1 + Y+ HX + HXY

A000045

A102426 triangle of Fibonacci parts

The number of binary strings of length n without consecutive Ys is the Fibonacci number Fn+2

1
••    •|    |•

Let an be strings of length n that start with 0, and bn n strings that starts with 1.Then:

an = an-1 + bn-1
bn = an-1 ; by summing we have:
an + bn = (an-1 + bn-1) + (an-2 + bn-2)
so we got the Fibonacci recurrence : Fn = Fn-1 + Fn-2

In terms of species,

F = X·F + X·G + X and
G = Y·F + Y = Y·X·F + Y·X·G + Y·X + Y that gives
F+G = (X + Y·X)·(F+G+1) + Y
By passing to e.g.f. for 1 + F + G we get:

G
# F
e
.
g
.
f
# .

1
y

1
−
x
−
x
y

{\displaystyle GF=e.g.f.={1+y \over 1-x-xy}}

## **Compositions with no singles** L(E–X–1)

A032032

```
1, 0, 1, 1, 7, 21, 141, 743, 5699, 42241,...

```

e
.
g
.
f
# .

1

2
x
−
e
x
p
(
x
)

{\displaystyle e.g.f.={1 \over 2+x-exp(x)}}

1 + E·F= F + F + X·F

0
2
3
2.2, 4
2.3, 3.2, 5
2.2.2, 2.4, 3.3, 4.2, 6
## **Odd compositions**, F = 1 + Odd.F

A006154 : 1 , 1, 2, 7, 32, 181, 1232, 9787, 88832,...

e
.
g
.
f
# .

1

1
−
s
i
n
h
(
x
)

{\displaystyle e.g.f.={\frac {1}{1-sinh(x)}}}

0
1
1.1
3, 1.1.1
3.1, 1.3, 1.1.1.1
5, 3.1.1, 1.3.1, 1.1.3, 1.1.1.1.1
Officer Miller

> "*The road will be open soon.*"
> ― Officer Miller
>
>

**Officer Miller** is a police officer who was introduced in S01E07.

## ***Biography***

### **S01E07**

Officer Miller along with several other officers and a bomb squad truck were at the town in Cadort Island investigating a potential bomb threat. Deminer Norman, one of the bomb squad officers tells the civilians to be patient as they have to make sure the bomb threat isn't real. Officer Miller replied that the road will open soon.

## ***Trivia***

* This character is not the same as Officer Miller who appeared in the movie series.
Math Symbols

a
n

x
n

G

{\displaystyle a\_n\frac {x^n} {|G|}}

## Fandom Latex

* latex-programming.fandom.com Align
* latex-programming.fandom.com
* latex.wikia.org
* Sets
* Tables

 zzz | zzz | zzz

## symbols

http://typefacts.com/keyboard-shortcuts
http://math.wikia.com/wiki/Greek\_alphabet

– em dash, long dash

B

k

→

B

{\displaystyle \mathcal{B}^k \rightarrow \mathcal{B}}

.

f

X

=
f

{\displaystyle f|\_{X}=f}

\mathbf {S}

## Wikipedia Latex

* https://en.wikipedia.org/wiki/Help:Wiki\_markup
* https://en.wikipedia.org/wiki/Help:Displaying\_a\_formula
* https://www.math-linux.com/latex-26/faq/latex-faq/article/horizontal-and-vertical-curly-braces-left-right-underbrace-and-overbrace

(

n
k

)

=

n
!

k
!
(
n
−
k
)
!

{\displaystyle {n\choose k}=\frac{n!}{k!(n-k)!}}

∑

# i
0

n
−
1

i

{\displaystyle \sum\_{i=0}^{n-1} i}
Lydia

> "*Hey, I got the presents and I'm on my way, I'll be there in 30-40 minutes. Just make sure not to ruin the surprise!*"
> ― Lydia
>
>

**Lydia** is a civilian character who was introduced in S02E07.

## ***Biography***

### **S02E07**

Lydia drives along the freeway in River Highway. She was on the phone telling someone that she will be at her location in 30-40 minutes with the presents and not to spoil the surprise. Later on at a crash site, Otto successfully flips a semi truck back on its wheels. However the truck starts to roll as the parking brake was not on. The truck rams Ophelia's minivan into the freeway and the drivers on the freeway crash into the minivan and the semi truck. Kimberly hits Lydia and Lydia swerves and crashes into the semi truck. A Gavril B-Series Flatbed crashes into Lydia. She survived the accident.
Classic PIE

## 3916900

https://math.stackexchange.com/questions/3916900/combinatorics-problem-choosing-6-cards-out-of-32-card-deck-so-that-there-a

There are four qualities that hands may have. There are hands that miss clubs, hands that have no hearts, hands without spades and hands without diamonds.

The situation is tricky because the binomial coefficients intervene in a double way. To reveal this let us use $N(x)$ and $E(x)$ the generating functions for the "hands having at least x qualities" and "hands having exactly x qualities".

Shortly, P.I.E says that $N(x)= E(x+1)$

Here

$N(x) = \binom 4 3 \binom 8 6 x^3 + \binom 4 2 \binom {16} 6 x^2 + \binom 4 1 \binom {24} 6x + \binom 4 0 \binom {32} 6$

$N(x) =112x^3 + 48048x^2 + 538384x + 906192$

$E(x) = N(x-1) = 12x^3 + 47712x^2 + 442624x + 415744$

As one may see, $442624$, the desired number, is

$442664 = \color {red}{\binom 3 1 }\binom 4 3 \binom 8 6 - \color{red}{\binom 2 1 } \binom 4 2 \binom {16} 6 + \color{red}{\binom 1 1 } \binom 4 1 \binom {24} 6$

Suppose now we have a 3-color deck of 24 cards and we are interested in all-color hands. Again,

$N(x) = \binom 3 2 \binom 8 6 x^2 + \binom 3 1 \binom {16} 6 x + \binom 3 0 \binom {24} 6 $

$N(x) =84x^2 + 24024x + 134596$

$E(x) = N(x-1) = 84x^2 + 23846x + 110656$

$110656 = \color {red}{\binom 2 0 }\binom 3 2 \binom 8 6 - \color{red}{\binom 1 0 } \binom 3 1 \binom {16} 6 + \color{red}{\binom 0 0 } \binom 3 0 \binom {24} 6$

Again, as we may see, $442624 = 4\times 110656$

E
x
a
c
t
(
t
# )
A
t
L
e
a
s
t
(
t
−
1
)

{\displaystyle Exact(t)=AtLeast(t-1)}

# -= Card decks =-

there are 13 + 4 sorts. There are "exact terms" only for k=2.

E
(
x
# )
52

x

2

{\displaystyle E(x)=52x^{2}}

N
(
x
# )
52
102
x
52

x

2

{\displaystyle N(x)=52+102x+52x^{2}}

the 102 "at least" terms are four aces + four kings + ... + thirteen spades + ... + thirteen clubs.

## Red and blue balls

3 red balls and 7 blue balls. There are three properties : ball, red, blue.

E
(
x
# )
10

x

2

{\displaystyle E(x)=10x^{2}}

N
(
x
# )
10
20
x
10

x

2

{\displaystyle N(x)=10+20x+10x^{2}}

https://math.stackexchange.com/questions/2939749/calculate-probability-using-inclusion-exclusion-and-deduce-formula-for-binomial calculate using inclusion-exclusion the probability that m special people are in the group

https://math.stackexchange.com/questions/2854867/calculating-the-probability-of-at-least-one-of-x-i-subset-x-using-inclusion-e

(

n
−
m

k
−
m

)

=

∑

# j
0

m

(
−
1

)

j

(

m
j

)

(

n
−
j

k

)

{\displaystyle {\begin{aligned}{n-m \choose k-m}=\sum \_{j=0}^{m}(-1)^{j}{m \choose j}{n-j \choose k}\end{aligned}}}
Formal Languages and Automata

## Formal languages and automata

S
→
λ

{\displaystyle S \rightarrow \lambda }

S
→
a
S
b

{\displaystyle S \rightarrow aSb }

a
n

b
n

,

n
⩾
0

{\displaystyle a^nb^n, \ n \geqslant 0 }

# S
1
a
S
b

{\displaystyle S = 1 + aSb }

# s

1

1
−
a
b

{\displaystyle s = { 1 \over 1- ab } }
 FL&A 23

A
→
λ

{\displaystyle A \rightarrow \lambda }

A
→
a
A
b

{\displaystyle A \rightarrow aAb }

S
→
A
b

{\displaystyle S \rightarrow Ab }

a
n

b

n
1

,

n
⩾
0

{\displaystyle a^nb^{n+1}, \ n \geqslant 0 }

# S
b
a
S
b

{\displaystyle S = b + aSb }

# s

b

1
−
a
b

{\displaystyle s = { b \over 1- ab } }
 FL&A 24

S
→
λ

{\displaystyle S \rightarrow \lambda }

S
→
S
S

{\displaystyle S \rightarrow SS }

S
→
a
S
b

{\displaystyle S \rightarrow aSb }

S
→
b
S
a

{\displaystyle S \rightarrow bSa }

n
b

(
w
# )

n
a

(
w
)

{\displaystyle n\_b(w) = n\_a(w) }

# S
1
2
a
b
S

S
2

{\displaystyle S = 1 + 2abS + S^2 }

  ambiguous

S
2

=
1
4
a
b

S
2

{\displaystyle S^2 = 1 + 4abS^2 }

1, 2, 6, 20, 70, 252, 924,

(

2
n

n

)

{\displaystyle {2n \choose n} }

1

1
−
4
a
b

{\displaystyle \frac 1 { \sqrt{1 - 4ab}} }

S
→
λ

{\displaystyle S \rightarrow \lambda }

S
→
a
S
b
S

{\displaystyle S \rightarrow aSbS }

S
→
b
S
a
S

{\displaystyle S \rightarrow bSaS }

n
b

(
w
# )

n
a

(
w
)

{\displaystyle n\_b(w) = n\_a(w) }

# S
1
2
a
b

S
2

{\displaystyle S = 1 + 2abS^2 }

  ambiguous

## Unambigous

S
→
λ

{\displaystyle S \rightarrow \lambda }

S
→
a
S
b
S

{\displaystyle S \rightarrow aSbS }

n
b

(
w
# )

n
a

(
w
)

{\displaystyle n\_b(w) = n\_a(w) }

# S
1
a
S
b
S

{\displaystyle S = 1 + aSbS }

1
−

1
−
4
a
b

2
a
b

{\displaystyle \frac {1- \sqrt{1 - 4ab}}{2ab} }
 Pascal

S
→
λ

{\displaystyle S \rightarrow \lambda }

S
→
a
S

{\displaystyle S \rightarrow aS }

S
→
b
S

{\displaystyle S \rightarrow bS }

∗

∗
∗

∗
∗

# S
1
a
S
b
S

{\displaystyle S = 1 + aS + bS }

(

a
b

a
,
b

)

{\displaystyle {a+b \choose a, b} }
 Fibonacci

S
→
λ

{\displaystyle S \rightarrow \lambda }

S
→
a
S

{\displaystyle S \rightarrow aS }

S
→
b

{\displaystyle S \rightarrow b }

S
→
b
a
S

{\displaystyle S \rightarrow baS }

# S
1
a
S
b
b
a
S

{\displaystyle S = 1 + aS + b + baS }

# s

1
b

1
−
a
−
b
a

{\displaystyle s = { 1 + b \over 1- a - ba } }

S
→
λ

{\displaystyle S \rightarrow \lambda }

S
→
a
S

{\displaystyle S \rightarrow aS }

S
→
b

a
∗

b
S

{\displaystyle S \rightarrow ba^\*bS }

# S
1
a
S
b

a
∗

b
S

{\displaystyle S = 1 + aS + ba^\*bS }

1
−
a

(
1
−
a

)
2

−

b
2

{\displaystyle \frac {1-a} { (1-a)^2 - b^2 } }