aarjaneiro/planetmath huggingface dataset:
Fibonacci jigsaw puzzle

Fibonacci jigsaw puzzle

In the

Fibonacci jigsaw puzzle

, one is asked to take a square of given area

x

2

and rearrange it into a

rectangle

of greater or lesser area by making the lengths correspond to

Fibonacci numbers

. Of course there is no

solution

, but the appearance of a valid solution can be created. The square with sides 8 units long (with an area of 64 square units) is the one most commonly used for presenting this purpose.

By cutting the plywood or paper (or whatever material one has used to make the square) as indicated by the green lines, one now has four pieces. Rotate the bottom two pieces 90 degrees in opposite directions. Slide the topmost piece to the right of one of the former bottom pieces, and the other former top piece to the left of one of the former bottom pieces. Most

presentations

of the puzzle then state that the rectangle has an area of 65 square units and ask how this is possible. The diagram normally shown has a

central

diagonal

that looks misleadingly

straight

. However, by making this puzzle out of an actual material and placing it on a surface with a differently-patterned surface, it becomes obvious that there is a gap in the middle of the rectangle.

The area of this gap turns out to be the 1 square unit that was seemingly created by rearranging the square into a rectangle.

Title

Fibonacci jigsaw puzzle

Canonical name

FibonacciJigsawPuzzle

Date of creation

2013-03-22 18:03:42

Last modified on

2013-03-22 18:03:42

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Result

Classification

msc 11B39

Generated on Fri Feb 9 15:27:34 2018 by

LaTeXML
Fibonacci sequence

Fibonacci sequence

The

Fibonacci sequence

, discovered by Leonardo Pisano Fibonacci, begins

0

,

1

,

1

,

2

,

3

,

5

,

8

,

13

,

21

,

34

,

55

,

89

,

144

,

233

,

377

,

…

(

Sequence

http://www.research.att.com/projects/OEIS?Anum=A000045

A000045 in

[

1

]

).
The

n

th

Fibonacci number

is generated by adding the previous two. Thus, the Fibonacci sequence has the

recurrence relation

f

n

=

f

n

-

1

+

f

n

-

2

with

f

0

=

0

and

f

1

=

1

. This recurrence relation can be solved into the

closed form

f

n

=

1

5

⁢

(

ϕ

n

-

ϕ

′

⁣

n

)

called the

Binet formula

, where

ϕ

denotes the

golden ratio

(and

ϕ

′

is defined in the same entry). Note that

lim

n

→

∞

⁡

f

n

+

1

f

n

=

ϕ

.

References

1

N. J. A. Sloane, (2004), The

On-Line Encyclopedia of Integer Sequences

,

http://www.research.att.com/ njas/sequences/

http://www.research.att.com/ njas/sequences/.

Title

Fibonacci sequence

Canonical

name

FibonacciSequence

Date of creation

2013-03-22 11:56:07

Last modified on

2013-03-22 11:56:07

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

21

Author

Koro (127)

Entry type

Definition

Classification

msc 11B39

Synonym

Fibonacci number

Related topic

HogattTheorem

Related topic

LucasNumbers

Related topic

ZeckendorfsTheorem

Related topic

ApplicationsOfSecondOrderRecurrenceRelationFormula

Defines

Binet formula

Generated on Fri Feb 9 15:27:36 2018 by

LaTeXML
Hogatt’s theorem

Hogatt’s theorem

Hogatt’s

theorem

states that every

positive integer

can be expressed as
a sum of distinct

Fibonacci numbers

.

For any positive integer,

k

∈

ℤ

+

, there exists a unique positive integer

n

so that

F

n

-

1

<

k

≤

F

n

. We proceed by strong

induction

on

n

. For

k

=

0

,

1

,

2

,

3

, the property is true as

0

,

1

,

2

,

3

are themselves Fibonacci numbers. Suppose

k

≥

4

and that every

integer

less than

k

is a sum of distinct Fibonacci numbers. Let

n

be the largest positive integer such that

F

n

<

k

. We first note that if

k

-

F

n

>

F

n

-

1

then

F

n

+

1

≥

k

>

F

n

+

F

n

-

1

=

F

n

+

1

,

giving us a

contradiction

. Hence

k

-

F

n

≤

F

n

-

1

and consequently the positive integer

(

k

-

F

n

)

can be expressed as a sum of distinct Fibonacci numbers. Moreover, this sum does not contain the term

F

n

as

k

-

F

n

≤

F

n

-

1

<

F

n

.
Hence,

k

=

(

k

-

F

n

)

+

F

n

is a sum of distinct Fibonacci numbers and Hogatt’s theorem is proved by induction.

Title

Hogatt’s theorem

Canonical name

HogattsTheorem

Date of creation

2013-03-22 13:43:24

Last modified on

2013-03-22 13:43:24

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 11B39

Related topic

FibonacciSequence

Generated on Fri Feb 9 15:27:38 2018 by

LaTeXML
Jacobsthal sequence

Jacobsthal sequence

The

Jacobsthal sequence

is an

additive

sequence

similar

to the

Fibonacci sequence

, defined by the

recurrence relation

J

n

=

J

n

-

1

+

2

⁢

J

n

-

2

, with initial terms

J

0

=

0

and

J

1

=

1

. A number in the sequence is called a

Jacobsthal number

. The first few are 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, etc., listed in A001045 of Sloane’s OEIS.

The

n

th Jacobsthal number is the

numerator

of the

alternating sum

∑

i

=

1

n

(

-

1

)

i

-

1

⁢

1

2

i

(the denominators are

powers of two

). This suggests a

closed form

: by putting the series solution over a common denominator and

summing

the geometric series in the numerator, we obtain two equations, one for even-indexed terms of the sequence,

J

2

⁢

n

=

2

2

⁢

n

-

1

3

and the other one for the odd-indexed terms,

J

2

⁢

n

+

1

=

2

2

⁢

n

+

1

-

2

3

+

1

.

These equations can be further generalized to

J

n

=

(

-

1

)

n

-

1

+

2

n

3

.

The Jacobsthal numbers are named after the German mathematician Ernst Jacobsthal.

Title

Jacobsthal sequence

Canonical name

JacobsthalSequence

Date of creation

2013-03-22 18:09:40

Last modified on

2013-03-22 18:09:40

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11B39

Defines

Jacobsthal number

Generated on Fri Feb 9 15:27:40 2018 by

LaTeXML
Lucas numbers

Lucas numbers

The

Lucas numbers

are a slight variation of

Fibonacci numbers

.
These numbers follow the same recursion:

l

n

+

1

=

l

n

+

l

n

-

1

but having different

initial conditions

:

l

1

=

1

,

l

2

=

3

leading to the

sequence

1

,

3

,

4

,

7

,

11

,

18

,

29

,

47

,

76

,

123

,

…

.

Lucas numbers have the following property:

l

n

=

f

n

-

1

+

f

n

+

1

where

f

n

is the

n

t

⁢

h

Fibonacci number.

Title

Lucas numbers

Canonical name

LucasNumbers

Date of creation

2013-03-22 12:19:58

Last modified on

2013-03-22 12:19:58

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 11B39

Synonym

Lucas sequence

Related topic

FibonacciSequence

Generated on Fri Feb 9 15:27:47 2018 by

LaTeXML
morphic number

morphic number

The

golden ratio

φ

=

1

+

5

2

satisfies

the equations

{

φ

+

1

=

φ

2

,

φ

-

1

=

φ

-

1

(1)

from which the latter is obained from the former by dividing by

φ

.
There is a pair of equations satisfied by the

plastic number

P

:

{

P

+

1

=

P

3

,

P

-

1

=

P

-

4

(2)

Here, the latter equation is justified by

P

5

-

P

4

-

1

≡

(

P

3

-

P

-

1

⏟

=

0

)

⁢

(

P

2

-

P

+

1

)

when this is divided by

P

4

.

An

algebraic integer

is called a

morphic number

, iff it satisfies a pair of equations

{

x

+

1

=

x

m

,

x

-

1

=

x

-

n

(3)

for some

positive integers

m

and

n

.

Accordingly, the golden ratio and the plastic number are morphic numbers.  It can be shown that there are no other real morphic numbers greater than 1.

References

1

J. Aarts, R. Fokkink, G. Kruijtzer

: Morphic numbers.  –

Nieuw Archief voor Wiskunde

5/2 (2001).

Title

morphic number

Canonical name

MorphicNumber

Date of creation

2013-03-22 19:09:51

Last modified on

2013-03-22 19:09:51

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Definition

Classification

msc 11B39

Generated on Fri Feb 9 15:27:49 2018 by

LaTeXML
Padovan sequence

Padovan sequence

Construct a

recurrence relation

with initial terms

a

0

=

1

,

a

1

=

0

,

a

2

=

0

and

a

n

=

a

n

-

3

+

a

n

-

2

for

n

>

2

. The first few terms of the

sequence

defined by this recurrence relation are: 1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151 (listed in A000931 of Sloane’s OEIS). This is the

Padovan sequence

, named after mathematician Richard Padovan. Its

generating function

is

G

⁢

(

a

⁢

(

n

)

;

x

)

=

1

-

x

2

1

-

x

2

-

x

3

.

It has been observed that in taking seven consecutive terms of this sequence, the sum of the squares of the first, third and seventh terms is equal to the sum of the squares of the second, fourth, fifth and sixth terms.

The

n

th Padovan number asymptotically matches the

n

th power of the

plastic constant

.

Title

Padovan sequence

Canonical name

PadovanSequence

Date of creation

2013-03-22 16:37:21

Last modified on

2013-03-22 16:37:21

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11B39

Generated on Fri Feb 9 15:27:50 2018 by

LaTeXML
Pell number

Pell number

A number in the

sequence

created from the

recurrence relation

P

n

=

2

⁢

P

n

-

1

+

P

n

-

2

,

with

P

0

=

0

and

P

1

=

1

.

The first few

Pell numbers

are 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, listed in A000129 of Sloane’s OEIS.

A Pell number for any given index can also be calculated from earlier Pell numbers with

P

a

+

b

=

P

a

⁢

P

b

+

1

+

P

a

-

1

⁢

P

b

.

The formula

-

(

1

-

2

)

n

+

(

1

+

2

)

n

2

⁢

2

works too. From this particular formula it can be deduced that the sequence of Pell numbers can be used in a

continued fraction

of the

square root of 2

as well as the silver ratio.

Yet another way to calculate Pell numbers is by squaring the terms of Pascal’s triangle and adding up the antidiagonals. Arranging the

Markov numbers

in a binary graph tree and reading the numbers on 2’s branch gives the Pell numbers with odd indices.

Only Pell numbers with prime indexes can also be prime. This fact is used in some tests for pseudoprimality.

Title

Pell number

Canonical name

PellNumber

Date of creation

2013-03-22 15:46:44

Last modified on

2013-03-22 15:46:44

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

6

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11B39

Defines

Pell number

Generated on Fri Feb 9 15:27:52 2018 by

LaTeXML
Perrin pseudoprime

Perrin pseudoprime

Given the

Perrin sequence

,

a

0

=

3

,

a

1

=

0

,

a

2

=

2

and

a

n

=

a

n

-

3

+

a

n

-

2

for

n

>

2

, if

n

is not prime yet

n

|

a

n

, that number is then called a

Perrin pseudoprime

. The first few Perrin pseudoprimes are 271441, 904631, 16532714, 24658561, 27422714, 27664033, 46672291 (listed in A013998 of Sloane’s OEIS).

By itself this is not a good enough test of

primality

. Further requiring that

a

n

≡

0

mod

n

and

a

-

n

≡

1

mod

n

improves the test but not enough to be preferable to actually

factoring

out the number. The first few

strong Perrin pseudoprimes

are 27664033, 46672291, 102690901, 130944133, 517697641 (listed in A018187 of Sloane’s OEIS).

Unlike Perrin pseudoprimes, most other

pseudoprimes

(

http://planetmath.org/PseudoprimeP

) are because of a

congruence relation

to a given base.

Title

Perrin pseudoprime

Canonical name

PerrinPseudoprime

Date of creation

2013-03-22 16:11:33

Last modified on

2013-03-22 16:11:33

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11B39

Defines

strong Perrin pseudoprime

Generated on Fri Feb 9 15:27:55 2018 by

LaTeXML
Perrin sequence

Perrin sequence

Construct a

recurrence relation

with initial terms

a

0

=

3

,

a

1

=

0

,

a

2

=

2

and

a

n

=

a

n

-

3

+

a

n

-

2

for

n

>

2

. The first few terms of the

sequence

defined by this recurrence relation are: 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, 51, 68, 90, 119, 158, 209, 277, 367 (listed in A001608 of Sloane’s OEIS). This is the

Perrin sequence

, sometimes called the

Ondrej Such sequence

. Its

generating function

is

G

⁢

(

a

⁢

(

n

)

;

x

)

=

3

-

x

2

1

-

x

2

-

x

3

.

A number in the Perrin sequence is called a

Perrin number

.

It has been observed that if

n

|

a

n

, then

n

is a

prime number

, at least among the first

hundred

thousand

integers

or so. However, the square of 521 passes this test.

The

n

th Perrin number asymptotically matches the

n

th power of the

plastic constant

.

References

1

W. W. Adams and D. Shanks, “Strong

primality tests

that are not

sufficient

”

Math. Comp.

39

, pp. 255 - 300 (1982)

Title

Perrin sequence

Canonical

name

PerrinSequence

Date of creation

2013-03-22 16:05:19

Last modified on

2013-03-22 16:05:19

Owner

Mravinci (12996)

Last modified by

Mravinci (12996)

Numerical id

5

Author

Mravinci (12996)

Entry type

Definition

Classification

msc 11B39

Defines

Perrin number

Generated on Fri Feb 9 15:27:57 2018 by

LaTeXML
plastic constant

plastic constant

Given the equation

P

3

=

P

+

1

, solve for

P

. The only solution in real numbers is

P

=

1

2

+

1

6

⁢

23

3

3

+

1

2

-

1

6

⁢

23

3

3

=

12

⁢

(

9

+

69

)

3

+

12

⁢

(

9

-

69

)

3

6

≈

1.3247179572447

, and

P

is the

plastic constant

, also known as the

silver number

.

Another way to calculate the plastic constant is

P

⁢

(

n

)

P

⁢

(

n

-

1

)

, where

P

⁢

(

n

)

is the

n

t

⁢

h

term of either the

Padovan sequence

or the

Perrin sequence

. For about

n

>

20

the approximation is adequate for all practical purposes.

Title

plastic constant

Canonical name

PlasticConstant

Date of creation

2013-03-22 16:10:39

Last modified on

2013-03-22 16:10:39

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11B39

Synonym

plastic number

Synonym

silver number

Synonym

silver constant

Generated on Fri Feb 9 15:27:58 2018 by

LaTeXML
primefree sequence

primefree sequence

Consider the

sequence

defined by

a

1

=

20615674205555510

,

a

2

=

3794765361567513

and

a

n

=

a

n

-

1

+

a

n

-

2

for all

n

>

2

. As it has been verified not to contain any primes, it is called a

primefree sequence

. The initial terms must be

coprime

, or else the lack of primes is a

trivial

consequence

of the initial terms sharing a

divisor

other than 1.

Any Fibonacci-like sequence will naturally exhibit some patterns in the factorizations of its terms in

relation

to their indices. The initial terms are chosen so that these patterns cover any possible value of

n

. So, for our example sequence, discovered by Wilf in 1990,

2

|

a

3

⁢

x

+

1

,

3

|

a

4

⁢

x

+

2

,

5

|

a

5

⁢

x

+

1

,

7

|

a

8

⁢

x

, etc. for a finite number of potential

prime factors

(and

x

≥

0

in each case).

Order is very important: switching the initial terms can cause primes to arise in the sequence. Switching the initial terms in our example causes

a

138

and a few others afterwards to be prime.

The example sequence is listed in A083216 of the OEIS.

References

1

P. Hoffman.

The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth

. New York: Hyperion, 1998.

2

H. Nicol. A Fibonacci-like sequence of

composite numbers

.

Electronic J. of

Combinatorics

6, 1999.

3

H. S. Wilf. Letters to the Editor.

Math. Mag.

63, 284, 1990.

Title

primefree sequence

Canonical name

PrimefreeSequence

Date of creation

2013-03-22 15:54:49

Last modified on

2013-03-22 15:54:49

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

7

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11B39

Generated on Fri Feb 9 15:28:01 2018 by

LaTeXML
proof of Cassini’s identity

proof of Cassini’s identity

For all

positive integers

i

, let

F

i

denote the

i

t

⁢

h

Fibonacci number

, with

F

1

=

F

2

=

1

. We will show by

induction

that the

identity

F

n

+

1

⁢

F

n

-

1

-

F

n

2

=

(

-

1

)

n

holds for all positive integers

n

≥

2

.
When

n

=

2

, we can substitute in the values for

F

1

,

F

2

and

F

3

yielding the statement

2

×

1

-

1

2

=

(

-

1

)

2

, which is true.
Now suppose that the

theorem

is true when

n

=

m

,
for some

integer

m

≥

2

.
Recalling the

recurrence relation

for the Fibonacci numbers,

F

i

+

1

=

F

i

+

F

i

-

1

, we have

F

m

+

2

⁢

F

m

-

F

m

+

1

2

=

(

F

m

+

1

+

F

m

)

⁢

F

m

-

(

F

m

+

F

m

-

1

)

2

=

F

m

+

1

⁢

F

m

+

F

m

2

-

F

m

2

-

2

⁢

F

m

⁢

F

m

-

1

-

F

m

-

1

2

=

F

m

+

1

⁢

F

m

-

2

⁢

F

m

⁢

F

m

-

1

-

F

m

-

1

2

=

(

F

m

+

F

m

-

1

)

⁢

F

m

-

2

⁢

F

m

⁢

F

m

-

1

-

F

m

-

1

2

=

F

m

2

+

F

m

-

1

⁢

F

m

-

2

⁢

F

m

⁢

F

m

-

1

-

F

m

-

1

2

=

F

m

2

-

F

m

⁢

F

m

-

1

-

F

m

-

1

2

=

F

m

2

-

(

F

m

+

F

m

-

1

)

⁢

F

m

-

1

=

F

m

2

-

F

m

+

1

⁢

F

m

-

1

=

-

(

-

1

)

m

by the

induction hypothesis

.
So we get

F

m

+

2

⁢

F

m

-

F

m

+

1

2

=

(

-

1

)

m

+

1

,
and the result is thus true for

n

=

m

+

1

.
The theorem now follows by induction.

Title

proof of Cassini’s identity

Canonical

name

ProofOfCassinisIdentity

Date of creation

2013-03-22 14:44:40

Last modified on

2013-03-22 14:44:40

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

24

Author

yark (2760)

Entry type

Proof

Classification

msc 11B39

Related topic

CatalansIdentity

Generated on Fri Feb 9 15:28:04 2018 by

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proof that a Zeckendorf representation represents a unique positive integer

proof that a Zeckendorf representation represents a unique positive integer

Theorem

.

For any

positive integer

n

, the

Zeckendorf representation

Z

(with

k

elements

all 0s or 1s) of

n

is unique.

For our proof, we accept it as axiomatic that

F

0

=

F

1

=

1

but

F

0

is not used in the Zeckendorf representation of any number, and we also accept as axiomatic that all

F

i

are distinct as long as

i

>

0

.

Proof.

Assume that there are two

integers

a

and

b

such that

0

<

a

<

b

, yet they both have the same Zeckendorf representation

Z

with

k

elements all 0s or 1s. We compute

c

=

∑

i

=

1

k

Z

i

⁢

F

i

,

where

F

x

is the

x

th

Fibonacci number

. We can be assured that there is only one possible value for

c

since all

F

i

are distinct for

i

>

0

and each

F

i

was added only once or not at all, since each

Z

i

is limited by definition to 0 or 1. Now

c

holds the value of the Zeckendorf representation

Z

. If

c

=

a

, it follows that

c

<

b

, but that would mean that

Z

is not the Zeckendorf representation of

b

after all, hence this results in a

contradiction

of our initial

assumption

. And if on the other hand

c

=

b

and

c

>

a

, then this leads to a

similar

contradiction as to what is the Zeckendorf representation of

a

really is.
∎

Title

proof that a Zeckendorf representation represents a unique positive integer

Canonical name

ProofThatAZeckendorfRepresentationRepresentsAUniquePositiveInteger

Date of creation

2013-03-22 16:36:46

Last modified on

2013-03-22 16:36:46

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Proof

Classification

msc 11B39

Classification

msc 11A63

Generated on Fri Feb 9 15:28:13 2018 by

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random Fibonacci sequence

random Fibonacci sequence

A

random Fibonacci sequence

has its initial terms defined as

F

0

=

F

1

=

1

but the sign (plus or minus) in

recurrence relation

F

n

=

F

n

-

1

±

F

n

-

2

is chosen randomly with either sign having an equal probability of being chosen. For example, if the random selection gives two minuses followed by three plusses, another minus, etc., the resulting random Fibonacci sequence is 1, 1, 0,

-

1

,

-

1

,

-

2

,

-

3

,

-

1

, etc. The scenarios that either plus or minus is always consistently chosen leads to the standard

Fibonacci sequence

, though in the latter case with all values (except the first two) multiplied by

-

1

.

In 2000, Divakar Viswanath proved that for most random Fibonacci sequences (with a few notable exceptions),

|

F

n

|

≈

⌊

V

n

⌋

, where

|

x

|

is the

absolute value

function

,

⌊

x

⌋

is the

floor function

and

V

is the

constant

1.13198824… now known as Viswanath’s constant. This does not hold true for the standard Fibonacci sequence nor for random Fibonacci sequences for which all

|

F

n

|

<

2

(such as 1, 1, 0, 1,

-

1

, 0,

-

1

, 1, 0, 1,

-

1

, generated by consistently alternating plus and minus at each turn), but for almost all other possible random Fibonacci sequences, “you can safely bet your life on the fact that for your

sequence

, the bigger

N

is, the closer the absolute value of the

N

th number gets to the

N

th power of 1.13198824…” (Devlin, 1999) In terms of limits,

lim

n

→

∞

⁡

|

F

n

|

n

=

V

.

References

1

Keith Devlin, “Devlin’s Angle: New Mathematical Constant Discovered” March 1999

MAA Online

2

Divakar Viswanath “Random Fibonacci sequences and the number 1.13198824….”

Mathematics of Computation

69

231 (2000): 1131 - 1155

Title

random Fibonacci sequence

Canonical name

RandomFibonacciSequence

Date of creation

2013-03-22 18:09:29

Last modified on

2013-03-22 18:09:29

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11B39

Generated on Fri Feb 9 15:28:15 2018 by

LaTeXML
Viswanath’s constant

Viswanath’s constant

Viswanath’s

constant

V

≈

1.1319882487943

is a

real number

whose

n

th power approximates the

absolute value

of the

n

th term of some

random Fibonacci sequences

, especially as

n

gets larger. In his 2000 paper, Divakar Viswanath gave the value of the

function

to just eight decimal places as 1.13198824. Viswanath believed the

logarithm

of the constant to lie between 0.123975598 and 0.1239755995. Oliveira and Figuereido in 2002 computed the value again using interval arithmetic instead of Viswanath’s “detailed rounding-error

analysis

,” in order to obtain “slightly better results.” Using Mathematica, Eric Weisstein computed a different value: 1.1321506910656020459.

The

continued fraction

of Viswanath’s constant, which is not periodic, begins

1

+

1

1

+

1

7

+

1

1

+

1

⋱

,

and aside from some

instances

of 2s, is thought to contain mostly

odd numbers

.

References

1

S. R. Finch,

Mathematical Constants

. Cambridge: Cambridge University Press (2003): 1.2.4

2

João Batista Oliveira & Luiz Henrique de Figuereido, “Interval Computation of Viswanath’s Constant”

Reliable Computing

8

2 (2002): 131 - 138

3

Divakar Viswanath “Random Fibonacci sequences and the number 1.13198824….”

Mathematics of Computation

69

231 (2000): 1131 - 1155

Title

Viswanath’s constant

Canonical name

ViswanathsConstant

Date of creation

2013-03-22 18:09:32

Last modified on

2013-03-22 18:09:32

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11B39

Synonym

Viswanath constant

Generated on Fri Feb 9 15:28:16 2018 by

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