OFFSET
0,4
COMMENTS
a(n+1) is the number of decimal digits of A001566(n).
From Hans J. H. Tuenter, Jul 24 2025, edited Feb 21 2026: (Start)
This sequence can be constructed by taking the first n digits of the binary expansion of
alpha = log_10(phi) = 0.00110 10110 00000 00011 ...
For example, expressing a(n)-1 in binary notation, gives
a(0)-1 = 0,
a(1)-1 = 0,
a(2)-1 = 0,
a(3)-1 = 1,
a(4)-1 = 11,
a(5)-1 = 110,
a(6)-1 = 1101,
a(7)-1 = 11010.
Another way of deriving the sequence is by the recurrence a(n+1)=2a(n)-1+d(n+1), with initial value a(0)=1, and d(n) the n-th digit in the binary expansion of alpha.
a(0) = 1,
a(1) = 2*1-1+0=1,
a(2) = 2*1-1+0=1,
a(3) = 2*1-1+1=2,
a(4) = 2*2-1+1=4,
a(5) = 2*4-1+0=7,
a(6) = 2*7-1+1=14,
a(7) = 2*14-1+0=27.
Alternatively, a(n) provides an encoding of the digits in the binary expansion of alpha,
d(n) = a(n)+1-2a(n-1) = 1-(a(n) mod 2), so that d(n) and a(n) have opposite parity.
One can also compute a(n) for a large value of n and fill in the sequence backward using a(n-1) = floor(a(n)/2) + (a(n) mod 2). This translates into a spigot-type algorithm to generate the binary digits of alpha from right to left. (End)
The sequence is defined recursively from the binary digits of log_10(phi), where phi = (1+sqrt(5))/2 is the golden ratio, rather than using the floor-based formulation. - Carlos M. da Fonseca, Apr 28 2026
LINKS
Hans J. H. Tuenter, Table of n, a(n) for n = 0..100
FORMULA
a(n) = 1+floor(2^n*log_10(phi)), where phi=(1+sqrt(5))/2, the golden ratio. - Hans J. H. Tuenter, Jul 23 2025
a(n) = 1 + Sum_{i=0..n} d(i)*2^(n-i), where d(i) is the i-th digit in the binary expansion of log_10(phi). - Hans J. H. Tuenter, Jul 24 2025
Sum_{n>=0} (a(n) mod 2)/2^n = 2 - log_10(phi). - Hans J. H. Tuenter, Feb 21 2026
EXAMPLE
a(5)=7, as L(2^5)=L(32)=4870847 and has seven digits.
MATHEMATICA
a[n_]:=1+Floor[2^n Log10[GoldenRatio]]; Table[a[n], {n, 0, 100}] (* Carlos M. da Fonseca, Feb 15 2026 *)
(* Alternative: *)
a = FoldList[2 #1 - 1 + #2 &, 1, RealDigits[N[Log[10, GoldenRatio], 100], 2, 100, -1][[1]]] (* Carlos M. da Fonseca, Apr 28 2026 *)
PROG
(PARI) a(n) = length(Str(fibonacci(2^(n+1))/fibonacci(2^n))); \\ adapted to new name by Michel Marcus, Jul 24 2025
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Matthijs Coster, Apr 29 2004
EXTENSIONS
More terms from Jason Earls, Apr 30 2004
a(23)-a(36) from Arkadiusz Wesolowski, Jul 20 2012
Name edited and a(0)=1 inserted by Hans J. H. Tuenter, Jul 23 2025
STATUS
approved