|
%I #16 Aug 09 2022 11:04:55
%S 1,6,62,618,6180,61804,618034,6180340,61803399,618033989,6180339888,
%T 61803398875,618033988750,6180339887499,61803398874990,
%U 618033988749895,6180339887498949,61803398874989485,618033988749894848,6180339887498948482,61803398874989484821
%N Number of 1's within a sample word of length 10^n of the infinite Fibonacci word A003842 where n is the sequence index.
%C The proportion of 1's within the sample word length tends to 1/phi = 0.6180339887... (A094214) as the sample size increases to infinity.
%H Rémy Sigrist, <a href="/A342959/b342959.txt">Table of n, a(n) for n = 0..1000</a>
%H Rémy Sigrist, <a href="/A342959/a342959.gp.txt">PARI program for A342959</a>
%F a(n) = A005206(10^n). - _Rémy Sigrist_, Apr 05 2021
%e 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.
%t set=Nest[Flatten[# /. {1 -> {1, 2}, 2 -> {1}}] &, {1}, 40]; Table[First@Counts@set[[1;;10^n]], {n, 1, 8}]
%o (PARI) See Links section.
%o (PARI) my(s=quadgen(5)-1); a(n) = floor((10^n+1)*s); \\ _Kevin Ryde_, Apr 09 2021
%o (Python)
%o from math import isqrt
%o def A342959(n): return ((m:=10**n+1)+isqrt(5*m**2)>>1)-m # _Chai Wah Wu_, Aug 09 2022
%Y Cf. A003842, A094214, A005614, A005206.
%K nonn
%O 0,2
%A _Frank M Jackson_, Mar 31 2021
%E a(0) = 1 prepended and more terms from _Rémy Sigrist_, Apr 05 2021
|
