A342959 Number of 1's within a sample word of length 10^n of the infinite Fibonacci word A003842 where n is the sequence index.

1, 6, 62, 618, 6180, 61804, 618034, 6180340, 61803399, 618033989, 6180339888, 61803398875, 618033988750, 6180339887499, 61803398874990, 618033988749895, 6180339887498949, 61803398874989485, 618033988749894848, 6180339887498948482, 61803398874989484821

Offset

0,2

Comments

The proportion of 1's within the sample word length tends to 1/phi = 0.6180339887... (A094214) as the sample size increases to infinity.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..1000

Rémy Sigrist, PARI program for A342959

Formula

a(n) = A005206(10^n). - Rémy Sigrist, Apr 05 2021

Example

a(1) = 6 because the first sample of the infinite Fibonacci word has a word length of 10. The word sample is (1, 2, 1, 1, 2, 1, 2, 1, 1, 2) and #1's = 6.

Mathematica

set=Nest[Flatten[# /. {1 -> {1, 2}, 2 -> {1}}] &, {1}, 40]; Table[First@Counts@set[[1;; 10^n]], {n, 1, 8}]

Prog

(PARI) See Links section.
(PARI) my(s=quadgen(5)-1); a(n) = floor((10^n+1)*s); \\ Kevin Ryde, Apr 09 2021
(Python)
from math import isqrt
def A342959(n): return ((m:=10**n+1)+isqrt(5*m**2)>>1)-m # Chai Wah Wu, Aug 09 2022

Crossrefs

Cf. A003842, A094214, A005614, A005206.

Sequence in context: A243643 A200803 A198965 * A144063 A186670 A190726

Adjacent sequences: A342956 A342957 A342958 * A342960 A342961 A342962

Keyword

nonn

Author

Frank M Jackson, Mar 31 2021

Extensions

a(0) = 1 prepended and more terms from Rémy Sigrist, Apr 05 2021

Status

approved