OpenLLM-France/wikimedia huggingface dataset:
# Cross product/Computation/C/2/Exercise

Compute the cross product
    ${\begin{pmatrix}4-7{\mathrm {i} }\\3+5{\mathrm {i} }\\-1-2{\mathrm {i} }\end{pmatrix}}\times {\begin{pmatrix}-6+4{\mathrm {i} }\\-2-8{\mathrm {i} }\\1-9{\mathrm {i} }\end{pmatrix}}$
in $\mathbb {C} ^{3}$.
# Cross product/Computation/Z mod 5/1/Exercise

Compute the cross product
    ${\begin{pmatrix}3\\1\\4\end{pmatrix}}\times {\begin{pmatrix}2\\4\\3\end{pmatrix}}$
in $K^{3}$, where $K$ denotes the field with five elements.
# Cross product/Computation/Z mod 7/1/Exercise

Compute the cross product
    ${\begin{pmatrix}4\\6\\1\end{pmatrix}}\times {\begin{pmatrix}5\\2\\5\end{pmatrix}}$
im $K^{3}$, where $K$ is the field with seven elements.
# Cross product/Bilinear/Alternating/Exercise

Let $K$ be a field. Show that the cross product on $K^{3}$ is bilinear and alternating.
# Cross product/Cyclic swapping/Exercise

Show that the cross product for vectors $x,y,z\in K^{3}$ fulfills the relation
    $x\times (y\times z)+y\times (z\times x)+z\times (x\times y)=0\,.$
# Linear algebra (Osnabrück 2024-2025)/Cross product/Cyclic swapping/Exercise/Exercisereferencenumber

Exercise 33.6
# Cross product/Orthonormal basis/Possibilities/Exercise

Let $u_{1},u_{2},u_{3}$ be an orthonormal basis of $\mathbb {R} ^{3}$. Show that $u_{1}\times u_{2}=\pm u_{3}$ holds.
# Field/Cross product/No basis/Exercise

Show that, over an arbitrary field $K$, for linearly independent vectors $u={\begin{pmatrix}a\\b\\c\end{pmatrix}}$ and $v={\begin{pmatrix}d\\e\\f\end{pmatrix}}$, the cross product ${\begin{pmatrix}a\\b\\c\end{pmatrix}}\times {\begin{pmatrix}d\\e\\f\end{pmatrix}}$ together with $u$ and $v$ do not necessarily form a basis of $K^{3}$.
# Isometries/R/Exercise

Determine the isometries on $\mathbb {R}$.
# Isometries/R^2/School/Exercise

What kind of isometries on $\mathbb {R} ^{2}$ do you know from school?
# Vector space/Inner product/Isometry/Injective/Exercise

Let $U,V$ denote ${\mathbb {K} }$-vector spaces, both endowed with an inner product, and let $\varphi \colon U\rightarrow V$ be an isometry. Show that $\varphi$ is injective.
# Vector space/Inner product/Isometry/Functorial properties/Exercise

Let $U,V,W$ be ${\mathbb {K} }$-vector spaces, each endowed with an inner product. Show the following statements.
a) The identity $V\rightarrow V$ is an isometry.
b) If $\varphi \colon U\rightarrow V$ is a bijective isometry, then the inverse mapping $\varphi ^{-1}$ is also an isometry.
c) If $\varphi \colon U\rightarrow V$ and $\psi \colon V\rightarrow W$ are isometries, then the composition $\psi \circ \varphi$ is also an isometry.
# Isometry/Several characterizations with orthonormal basis/Fact/Proof/Exercise

Let $V$ and $W$ be Euclidean vector spaces, and let
    $\varphi \colon V\longrightarrow W$
denote a linear mapping. Show that the following statements are equivalent.
* $\varphi$ is an isometry.
* For every orthonormal basis $u_{i}$, $i=1,\ldots ,n$, of $V$, $\varphi (u_{i})$, $i=1,\ldots ,n$, is part of an orthonormal basis of $W$.
* There exists an orthonormal basis $u_{i}$, $i=1,\ldots ,n$, of $V$ such that $\varphi (u_{i})$, $i=1,\ldots ,n$, is part of an orthonormal basis of $W$.
# Linear algebra (Osnabrück 2024-2025)/Isometry/Several characterizations with orthonormal basis/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 33.16
# Complex vector spaces/Inner product/Isometry/Real/Exercise

Let $V,W$ denote complex vector spaces, each endowed with an inner product, and let
    $\varphi \colon V\longrightarrow W$
denote a linear mapping. Show that $\varphi$ is an isometry with respect to the given complex inner products if and only if $\varphi$ is an isometry with respect to the corresponding real inner products.
# Proper isometry/Trigonalizable/Diagonalizable/Exercise

Let
    $f\colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{n}$
be a proper isometry. Suppose that $f$ is trigonalizable. Show that $f$ is already diagonalizable.
# Isometry/Euclidean vector space/Group/Exercise

Let $V$ be a Euclidean vector space. Show that the set of all isometries on $V$ forms a group under the composition of mappings.
# Cross product/Computation/Z mod 7/2/Exercise

Compute the cross product
    ${\begin{pmatrix}3\\2\\5\end{pmatrix}}\times {\begin{pmatrix}4\\6\\2\end{pmatrix}}$
in $K^{3}$, where $K$ denotes the field with seven elements.
# Isometry/Same space/Characterizations with orthonormal basis/Exercise

Let $V$ be a Euclidean vector space, and let
    $\varphi \colon V\longrightarrow V$
denote a linear mapping. Show that the following statements are equivalent.
* $\varphi$ is an isometry.
* For every vector $v$ with $\Vert {v}\Vert =1$, we also have $\Vert {\varphi (v)}\Vert =1$.
* For every orthonormal basis $u_{i},i=1,\ldots ,n$, also $\varphi (u_{i}),i=1,\ldots ,n$, is an orthonormal basis.
* There exists an orthonormal basis $u_{i},i=1,\ldots ,n$, such that also $\varphi (u_{i}),i=1,\ldots ,n$, is an orthonormal basis.
# Properties of Boolean functions/hard/binary

Properties of Boolean functions :
| hard | soft |
| binary | binary |
| integer | integer |
Hard properties can be assigned to a BF, without referencing its arity. Binary properties have the values true or false, so they partition all BF into a subset and the rest.

## linear

Linear Boolean functions are Walsh functions and their complements.

## seal

See Seal (discrete mathematics). Seal block is a broader propery.

## dense

no gaps before or between the atoms, i.e. valency = adicity (often called nondegenerate)

## strong

BF is strong, iff strength = adicity, so its family has the biggest possible size 2$^{adicity}$.

## balanced

same number of true and false places, i.e. weight = 0.5 BF that are not balanced are light or heavy, i.e. their weight is below or above 0.5.

## introverted

introverted iff complement is in same family, otherwise extroverted

## monotonic

no true place under false place (when places are arranged in a Hasse diagram) (counted by Dedekind numbers)

## related to parity

quadrants :
| even evil (0) | even odious (2) |
| odd evil (1) | odd odious (3) |
* parity: BF is odd/even iff first place in truth table is true/false. Same as parity of the Zhegalkin index.
* depravity: BF is odious/evil iff last place in truth table is true/false. Same as parity of the binary weight of the Zhegalkin index.
* BF is ugly, iff parity and depravity are different (i.e. iff the quadrant is 1 or 2).
* quadrant = parity + 2 · depravity
images for arity 3 :
| | odd | odious | ugly | quadrants | |
| truth tables | 1 1000 0000 | 128 0000 0001 | 129 1000 0001 | | Odd, odious and ugly are the three hard foibles. |
| Zhegalkin indices | 1 1000 0000 | 255 1111 1111 | 254 0111 1111 | | Odd, odious and ugly are the three hard foibles. |
* gender: BF is male/female, iff its root is sharp/blunt. See Gender of Boolean functions.

## honesty

BF is honest/dishonest, iff the XOR of all members of its family is the tautology/contradiction. (Most BF are honest.)
# Invertible matrices/K/Group/Exercise

Let $K$ be a field, and $n\in \mathbb {N}$. Show that the set $\operatorname {GL} _{n}\!{\left(K\right)}$ of invertible matrices is a group. Moreover, show that this group is, for $n\geq 2$, not commutative.
# Properties of Boolean functions

Properties of Boolean functions :
| hard | soft |
| binary | binary |
| integer | integer |
Properties of Boolean functions are maps that assign values to Boolean functions.
Soft properties are dependent on arity. For them the domain is always just the set of $2^{2^{n}}$ Boolean functions with arity $n$.
Hard properties are rarer, and probably more useful. For them the domain can be the infinite set of all Boolean functions. (Typically the codomain will also be infinite, unless the values are Boolean.)
Properties of 3-ary BF are illustrated here: Boolf prop/3-ary E.g. Boolf prop/3-ary/quaestor, Boolf prop/3-ary/patron
Below is a list of links to the respective chapters. They are created with {{boolf-prop}}.

## hard

* weight
* nonlinearity
* atoms
* root
* family, faction, clan
* super-family, super-faction, super-clan
* prefect
* legion, cohort
* complement
* reverse
* dual

### binary

* linear
* seal, seal block
* balanced, light, heavy
* introverted, extroverted
* dense
* [[|strong]]
* monotonic
* parity, odd, even
* depravity, odious, evil
* ugly
* quadrant
* gender, male, female
* honesty, honest, dishonest

### integer

* valency
* adicity
* arity
* strength
* gravity, depth

## soft

* twin
* half-complement
* ultra-family, ultra-clan
* tribe
* principality, dominion

### binary

* sharp, blunt
* rude
* rough
* acute, obtuse
* noble
* symmetric

### integer

* soft weight
* soft nonlinearity
* soft gravity, soft depth
* consul, signed consul, consul weight
* patron
* praetor
* quaestor
# Linear mapping/R^2/Order 2/No isometry/Exercise

Give an example of a linear mapping
    $\varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2}$
that is not an isometry and such that its order is $2$.
# Area-preserving/Not linear isometry/Example/Exercise

Give an example of a linear mapping
    $\varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2}$
such that $\varphi$ is area-preserving but not an isometry.
# Sequence Aloe

A051502
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 1 | 2 | 23 | 3904 | 134156284 | 288230371925149328 | 2658455991569831727504985413859223552 |
Oak ∘ Aloe = Olive
Oak ∘ truncated Aloe = Forsythia
Aloe ∘ powers of two = Lily
# Sequence Lily

| | valency gravity | strength depth |
| triangle | Willow | Poplar |
| diagonal | Lotus | Lily |
| difference | Orchid | Oxalis |

## Sequence Lily$_{(Drop)}$

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 Lily | 2 | 2 | 8 | 184 | 62464 | 4293001088 | 18446743803209556992 | 340282366920938461120638132973980614656 |
| 💧 LilyDrop | 2 | 0 | 6 | 176 | 62280 | 4292938624 | 18446743798916555904 | 340282366920938461102191389170771057664 |
Lily is the product of Aloe and the powers of two. :
| 0 | 2 | 2 · 1 |
| 1 | 2 | 1 · 2 |
| 2 | 8 | 2 · 4 |
| 3 | 184 | 23 · 8 |
| 4 | 62464 | 3904 · 16 |
| 5 | 4293001088 | 134156284 · 32 |
| 6 | 18446743803209556992 | 288230371925149328 · 64 |
| 7 | 340282366920938461120638132973980614656 | 2658455991569831727504985413859223552 · 128 |

## Sequence SmallLily$_{(Drop)}$

SmallLily differs from Lily only in the first entry. SmallLily$_{(Drop)}$ are the left columns of SmallTurnedPoplar$_{(Drop)}$.
| | 0 | 1 | 2 | 3 | 4 |
| 🌊 SmallLily (A305737) | 1 | 2 | 8 | 184 | 62464 |
| 💧 SmallLilyDrop | 1 | 1 | 6 | 176 | 62280 |
row sums of SignedCypress :
SignedCypress    row sums SmallLily (A305737) :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | sums |
| 0 | 1 · 1 1 | | | | | | | | 1 |
| 1 | 1 · 1 −1 | 1 · 3 3 | | | | | | | 2 |
| 2 | 2 · 1 2 | 3 · 3 −9 | 1 · 15 15 | | | | | | 8 |
| 3 | 8 · 1 −8 | 14 · 3 42 | 7 · 15 −105 | 1 · 255 255 | | | | | 184 |
| 4 | 64 · 1 64 | 120 · 3 −360 | 70 · 15 1050 | 15 · 255 −3825 | 1 · 65535 65535 | | | | 62464 |
| 5 | 1024 · 1 −1024 | 1984 · 3 5952 | 1240 · 15 −18600 | 310 · 255 79050 | 31 · 65535 −2031585 | 1 · 4294967295 4294967295 | | | 4293001088 |
| 6 | 32768 · 1 32768 | 64512 · 3 −193536 | 41664 · 15 624960 | 11160 · 255 −2845800 | 1302 · 65535 85326570 | 63 · 4294967295 −270582939585 | 1 · 18446744073709551615 18446744073709551615 | | 18446743803209556992 |
| 7 | 2097152 · 1 −2097152 | 4161536 · 3 12484608 | 2731008 · 15 −40965120 | 755904 · 255 192755520 | 94488 · 65535 −6192271080 | 5334 · 4294967295 22909355551530 | 127 · 18446744073709551615 −2342736497361113055105 | 1 · 340282366920938463463374607431768211455 340282366920938463463374607431768211455 | 340282366920938461120638132973980614656 |
# Euclidean vector space/Automorphism/Preserves orthogonality, no isometry/Exercise

Give an example of a bijective linear mapping
    $\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3}$
that is not an isometry but fulfills for all $u,v\in V$ the relation
    $\left\langle u,v\right\rangle =0{\text{ if and only if }}\left\langle \varphi (u),\varphi (v)\right\rangle =0.$
# Isometry/Orthogonal bijective determinant/Exercise

Let $V$ be a Euclidean vector space. Let
    $\varphi \colon V\longrightarrow V$
denote a linear mapping with the property that the determinant of $\varphi$ equals $1$ or $-1$. Moreover $\varphi$, satisfies the property that orthogonal vectors are mapped to orthogonal vectors. Show that $\varphi$ is an isometry.
# Linear mapping/R^2/Orthogonal/Orthonormal/Exercise

Give an example of a bijective linear mapping
    $\varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2}$
with the property that, on one hand, there exists an orthogonal basis of $\mathbb {R} ^{2}$ that is mapped under $\varphi$ to an orthogonal basis, and, on the other hand, there exists an orthogonal basis that is not mapped to an orthogonal basis.
# Isometries/C/Exercise

Determine the isometries on $\mathbb {C}$.
# Sequence Oxalis

| | valency gravity | strength depth |
| triangle | Willow | Poplar |
| diagonal | Lotus | Lily |
| difference | Orchid | Oxalis |

## Sequences Oxalis$_{(Drop)}$

These are the row sums of TurnedPoplar$_{(Drop)}$ without column 0.
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 Oxalis | 0 | 2 | 8 | 72 | 3072 | 1966208 | 270499994624 | 2342736474457787596800 |
| 💧 OxalisDrop | 0 | 2 | 6 | 64 | 3000 | 1963136 | 270498028416 | 2342736474187287602176 |
Oxalis is the difference of Grass and Lily. :
| | Oxalis | Grass − Lily |
| 0 | 0 | 2 − 2 |
| 1 | 2 | 4 − 2 |
| 2 | 8 | 16 − 8 |
| 3 | 72 | 256 − 184 |
| 4 | 3072 | 65536 − 62464 |
| 5 | 1966208 | 4294967296 − 4293001088 |
| 6 | 270499994624 | 18446744073709551616 − 18446743803209556992 |
| 7 | 2342736474457787596800 | 340282366920938463463374607431768211456 − 340282366920938461120638132973980614656 |
half :
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 HalfOxalis | 0 | 1 | 4 | 36 | 1536 | 983104 | 135249997312 | 1171368237228893798400 |
| 💧 HalfOxalisDrop | 0 | 1 | 3 | 32 | 1500 | 981568 | 135249014208 | 1171368237093643801088 |

## Sequences SmallOxalis$_{(Drop)}$

These are the row sums of SmallTurnedPoplar$_{(Drop)}$ without column 0.
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 SmallOxalis (A218384) | 0 | 1 | 7 | 71 | 3071 | 1966207 | 270499994623 | 2342736474457787596799 |
| 💧 SmallOxalisDrop | 0 | 1 | 6 | 64 | 3000 | 1963136 | 270498028416 | 2342736474187287602176 |
# Euclidean vector space/Orthonormal basis and isometry with R^n/Exercise

Let $V$ be a Euclidean vector space of dimension $n$. Show that a family of vectors $u_{1},\ldots ,u_{n}\in V$ is an orthonormal basis of $V$ if and only if the corresponding linear mapping
    $\mathbb {R} ^{n}\longrightarrow V,e_{i}\longmapsto u_{i},$
is an isometry between $\mathbb {R} ^{n}$ and $V$.
# Sequence Violet

| | diagonal | triangle | row sums |
| ♀ | Violet | Robinia | Primula |
| ♂ | HalfGrass | HalfCedar | HalfCrocus |
| all | Lotus | Willow | Grass |
| ♂ − ♀ | Orchid | Hickory | Hyacinth |
Violet is the difference of Lotus and HalfGrass:  ${\text{Violet}}(n)={\color {Green}{\text{Lotus}}(n)}-{\color {Red}2^{2^{n}-1}}$
| 0 | 1 | 2 − 1 |
| 1 | 0 | 2 − 2 |
| 2 | 2 | 10 − 8 |
| 3 | 90 | 218 − 128 |
| 4 | 31826 | 64594 − 32768 |
| 5 | 2147158386 | 4294642034 − 2147483648 |
| 6 | 9223372011085950170 | 18446744047940725978 − 9223372036854775808 |
| 7 | 170141183460469231602560095290109272522 | 340282366920938463334247399005993378250 − 170141183460469231731687303715884105728 |
All entries except the first are even. Their halves are A007537.
Violet is the diagonal of the triangles Robinia$_{(Drop)}$.
# Sequence Primula

| | diagonal | triangle | row sums |
| ♀ | Violet | Robinia | Primula |
| ♂ | HalfGrass | HalfCedar | HalfCrocus |
| all | Lotus | Willow | Grass |
| ♂ − ♀ | Orchid | Hickory | Hyacinth |
Primula = HalfCrocus − Hyacinth

## Sequences Primula$_{(Drop)}$

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 Primula (A246537) | 1 | 1 | 3 | 97 | 32199 | 2147318437 | 9223372023969379707 | 170141183460469231667123699412802366921 |
| 💧 PrimulaDrop | 1 | 0 | 2 | 94 | 32102 | 2147286238 | 9223372021822061270 | 170141183460469231657900327388832987214 |
row sums of Robinia$_{(Drop)}$ :
🌊    triangle Robinia    row sums Primula (A246537) :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 1 1 | | | | | | 1 |
| 1 | 1 · 1 1 | 1 · 0 0 | | | | | 1 |
| 2 | 1 · 1 1 | 2 · 0 0 | 1 · 2 2 | | | | 3 |
| 3 | 1 · 1 1 | 3 · 0 0 | 3 · 2 6 | 1 · 90 90 | | | 97 |
| 4 | 1 · 1 1 | 4 · 0 0 | 6 · 2 12 | 4 · 90 360 | 1 · 31826 31826 | | 32199 |
| 5 | 1 · 1 1 | 5 · 0 0 | 10 · 2 20 | 10 · 90 900 | 5 · 31826 159130 | 1 · 2147158386 2147158386 | 2147318437 |
💧    triangle RobiniaDrop    row sums PrimulaDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 1 1 | | | | | | 1 |
| 1 | 0 | 1 · 0 0 | | | | | 0 |
| 2 | 0 | 1 · 0 0 | 1 · 2 2 | | | | 2 |
| 3 | 0 | 1 · 0 0 | 2 · 2 4 | 1 · 90 90 | | | 94 |
| 4 | 0 | 1 · 0 0 | 3 · 2 6 | 3 · 90 270 | 1 · 31826 31826 | | 32102 |
| 5 | 0 | 1 · 0 0 | 4 · 2 8 | 6 · 90 540 | 4 · 31826 127304 | 1 · 2147158386 2147158386 | 2147286238 |
# Sequence Grass

| | diagonal | triangle | row sums |
| ♀ | Violet | Robinia | Primula |
| ♂ | HalfGrass | HalfCedar | HalfCrocus |
| all | Lotus | Willow | Grass |
| ♂ − ♀ | Orchid | Hickory | Hyacinth |

## Sequences Grass$_{(Drop)}$

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| 🌊 Grass (A001146) | 2 | 4 | 16 | 256 | 65536 | 4294967296 | 18446744073709551616 |
| 💧 GrassDrop (~A111403) | 2 | 2 | 12 | 240 | 65280 | 4294901760 | 18446744069414584320 |
half :
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| 🌊 HalfGrass (A058891) | 1 | 2 | 8 | 128 | 32768 | 2147483648 | 9223372036854775808 |
| 💧 HalfGrassDrop | 1 | 1 | 6 | 120 | 32640 | 2147450880 | 9223372034707292160 |

## Sequences SmallGrass$_{(Drop)}$

SmallGrass = Grass − 1 SmallGrassDrop is like GrassDrop, but the first entry is 1 instead of 2.
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| 🌊 SmallGrass (A051179) | 1 | 3 | 15 | 255 | 65535 | 4294967295 | 18446744073709551615 |
| 💧 SmallGrassDrop (~A111403) | 1 | 2 | 12 | 240 | 65280 | 4294901760 | 18446744069414584320 |
# Sequence Crocus

| | diagonal | triangle | row sums |
| ♀ | Violet | Robinia | Primula |
| ♂ | HalfGrass | HalfCedar | HalfCrocus |
| all | Lotus | Willow | Grass |
| ♂ − ♀ | Orchid | Hickory | Hyacinth |
Crocus = Hyacinth + Grass HalfCrocus = Hyacinth + Primula

## Sequences Crocus$_{(Drop)}$

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 Crocus (2·A246418) | 2 | 6 | 26 | 318 | 66674 | 4295297718 | 18446744099480343818 | 340282366920938463592501816037931689070 |
| 💧 CrocusDrop | 2 | 4 | 20 | 292 | 66356 | 4295231044 | 18446744095185046100 | 340282366920938463574055071938451345252 |
half :
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 HalfCrocus (A246418) | 1 | 3 | 13 | 159 | 33337 | 2147648859 | 9223372049740171909 | 170141183460469231796250908018965844535 |
| 💧 HalfCrocusDrop | 1 | 2 | 10 | 146 | 33178 | 2147615522 | 9223372047592523050 | 170141183460469231787027535969225672626 |
row sums of Cedar$_{(Drop)}$ :
🌊    triangle Cedar    row sums Crocus :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 2 2 | | | | | | 2 |
| 1 | 1 · 2 2 | 1 · 4 4 | | | | | 6 |
| 2 | 1 · 2 2 | 2 · 4 8 | 1 · 16 16 | | | | 26 |
| 3 | 1 · 2 2 | 3 · 4 12 | 3 · 16 48 | 1 · 256 256 | | | 318 |
| 4 | 1 · 2 2 | 4 · 4 16 | 6 · 16 96 | 4 · 256 1024 | 1 · 65536 65536 | | 66674 |
| 5 | 1 · 2 2 | 5 · 4 20 | 10 · 16 160 | 10 · 256 2560 | 5 · 65536 327680 | 1 · 4294967296 4294967296 | 4295297718 |
💧    triangle CedarDrop    row sums CrocusDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 2 2 | | | | | | 2 |
| 1 | 0 | 1 · 4 4 | | | | | 4 |
| 2 | 0 | 1 · 4 4 | 1 · 16 16 | | | | 20 |
| 3 | 0 | 1 · 4 4 | 2 · 16 32 | 1 · 256 256 | | | 292 |
| 4 | 0 | 1 · 4 4 | 3 · 16 48 | 3 · 256 768 | 1 · 65536 65536 | | 66356 |
| 5 | 0 | 1 · 4 4 | 4 · 16 64 | 6 · 256 1536 | 4 · 65536 262144 | 1 · 4294967296 4294967296 | 4295231044 |
half :
🌊    triangle HalfCedar    row sums HalfCrocus (A246418) :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 1 1 | | | | | | 1 |
| 1 | 1 · 1 1 | 1 · 2 2 | | | | | 3 |
| 2 | 1 · 1 1 | 2 · 2 4 | 1 · 8 8 | | | | 13 |
| 3 | 1 · 1 1 | 3 · 2 6 | 3 · 8 24 | 1 · 128 128 | | | 159 |
| 4 | 1 · 1 1 | 4 · 2 8 | 6 · 8 48 | 4 · 128 512 | 1 · 32768 32768 | | 33337 |
| 5 | 1 · 1 1 | 5 · 2 10 | 10 · 8 80 | 10 · 128 1280 | 5 · 32768 163840 | 1 · 2147483648 2147483648 | 2147648859 |
💧    triangle HalfCedarDrop    row sums HalfCrocusDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 1 1 | | | | | | 1 |
| 1 | 0 | 1 · 2 2 | | | | | 2 |
| 2 | 0 | 1 · 2 2 | 1 · 8 8 | | | | 10 |
| 3 | 0 | 1 · 2 2 | 2 · 8 16 | 1 · 128 128 | | | 146 |
| 4 | 0 | 1 · 2 2 | 3 · 8 24 | 3 · 128 384 | 1 · 32768 32768 | | 33178 |
| 5 | 0 | 1 · 2 2 | 4 · 8 32 | 6 · 128 768 | 4 · 32768 131072 | 1 · 2147483648 2147483648 | 2147615522 |
# Sequence Hyacinth

| | diagonal | triangle | row sums |
| ♀ | Violet | Robinia | Primula |
| ♂ | HalfGrass | HalfCedar | HalfCrocus |
| all | Lotus | Willow | Grass |
| ♂ − ♀ | Orchid | Hickory | Hyacinth |

## Sequences Hyacinth$_{(Drop)}$

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 Hyacinth (2·A059301) | 0 | 2 | 10 | 62 | 1138 | 330422 | 25770792202 | 129127208606163477614 |
| 💧 HyacinthDrop | 0 | 2 | 8 | 52 | 1076 | 329284 | 25770461780 | 129127208580392685412 |
Hyacinth is the difference of Crocus and Grass.   (Also of HalfCrocus and Primula.) :
| | Hyacinth | Crocus − Grass | HalfCrocus − Primula |
| 0 | 0 | 2 − 2 | 1 − 1 |
| 1 | 2 | 6 − 4 | 3 − 1 |
| 2 | 10 | 26 − 16 | 13 − 3 |
| 3 | 62 | 318 − 256 | 159 − 97 |
| 4 | 1138 | 66674 − 65536 | 33337 − 32199 |
| 5 | 330422 | 4295297718 − 4294967296 | 2147648859 − 2147318437 |
| 6 | 25770792202 | 18446744099480343818 − 18446744073709551616 | 9223372049740171909 − 9223372023969379707 |
| 7 | 129127208606163477614 | 340282366920938463592501816037931689070 − 340282366920938463463374607431768211456 | 170141183460469231796250908018965844535 − 170141183460469231667123699412802366921 |
half :
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 HalfHyacinth (A059301) | 0 | 1 | 5 | 31 | 569 | 165211 | 12885396101 | 64563604303081738807 |
| 💧 HalfHyacinthDrop | 0 | 2 | 4 | 26 | 538 | 164642 | 12885230890 | 64563604290196342706 |
row sums of Hickory$_{(Drop)}$ :
🌊    triangle Hickory    row sums Hyacinth (2·A059301) :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 0 0 | | | | | | 0 |
| 1 | 1 · 0 0 | 1 · 2 2 | | | | | 2 |
| 2 | 1 · 0 0 | 2 · 2 4 | 1 · 6 6 | | | | 10 |
| 3 | 1 · 0 0 | 3 · 2 6 | 3 · 6 18 | 1 · 38 38 | | | 62 |
| 4 | 1 · 0 0 | 4 · 2 8 | 6 · 6 36 | 4 · 38 152 | 1 · 942 942 | | 1138 |
| 5 | 1 · 0 0 | 5 · 2 10 | 10 · 6 60 | 10 · 38 380 | 5 · 942 4710 | 1 · 325262 325262 | 330422 |
💧    triangle HickoryDrop    row sums HyacinthDrop :
| kn | 0 | 1 | 2 | 3 | 4 | 5 | sums |
| 0 | 1 · 0 0 | | | | | | 0 |
| 1 | 0 | 1 · 2 2 | | | | | 2 |
| 2 | 0 | 1 · 2 2 | 1 · 6 6 | | | | 8 |
| 3 | 0 | 1 · 2 2 | 2 · 6 12 | 1 · 38 38 | | | 52 |
| 4 | 0 | 1 · 2 2 | 3 · 6 18 | 3 · 38 114 | 1 · 942 942 | | 1076 |
| 5 | 0 | 1 · 2 2 | 4 · 6 24 | 6 · 38 228 | 4 · 942 3768 | 1 · 325262 325262 | 329284 |