cp's OEIS Frontend

A094057 Number of decimal digits of Lucas(2^n).

%I A094057 #33 Jul 29 2025 18:46:11
%S A094057 1,1,1,2,4,7,14,27,54,108,215,429,857,1713,3425,6849,13697,27393,
%T A094057 54785,109570,219140,438279,876558,1753116,3506231,7012462,14024924,
%U A094057 28049847,56099693,112199386,224398771,448797541,897595081,1795190161,3590380321,7180760642,14361521283
%N A094057 Number of decimal digits of Lucas(2^n).
%C A094057 a(n+1) is the number of decimal digits of A001566(n).
%C A094057 From _Hans J. H. Tuenter_, Jul 24 2025: (Start)
%C A094057 This sequence can be constructed by taking the first n digits of the binary expansion of
%C A094057 alpha = log_10(phi) = 0.00110 10110 00000 00011 ...
%C A094057 For example, expressing a(n)-1 in binary notation, gives
%C A094057  a(0)-1 = 0,
%C A094057  a(1)-1 = 0,
%C A094057  a(2)-1 = 0,
%C A094057  a(3)-1 = 1,
%C A094057  a(4)-1 = 11,
%C A094057  a(5)-1 = 110,
%C A094057  a(6)-1 = 1101,
%C A094057  a(7)-1 = 11010.
%C A094057 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.
%C A094057  a(0) = 1,
%C A094057  a(1) = 2*1-1+0=1,
%C A094057  a(2) = 2*1-1+0=1,
%C A094057  a(3) = 2*1-1+1=2,
%C A094057  a(4) = 2*2-1+1=4,
%C A094057  a(5) = 2*4-1+0=7,
%C A094057  a(6) = 2*7-1+1=14,
%C A094057  a(7) = 2*14-1+0=27.
%C A094057 Alternatively, a(n) provides a encoding of the digits in the binary expansion of alpha,
%C A094057  d(n) = a(n)+1-2a(n-1). (End)
%H A094057 Hans J. H. Tuenter, <a href="/A094057/b094057.txt">Table of n, a(n) for n = 0..100</a>
%F A094057 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
%F A094057 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
%e A094057 a(5)=7, as L(2^5)=L(32)=4870847 and has seven digits.
%o A094057 (PARI) a(n) = length(Str(fibonacci(2^(n+1))/fibonacci(2^n))); \\ adapted to new name by _Michel Marcus_, Jul 24 2025
%Y A094057 Cf. A000032, A001566, A001622, A055642, A114469.
%K A094057 nonn,base
%O A094057 0,4
%A A094057 _Matthijs Coster_, Apr 29 2004
%E A094057 More terms from _Jason Earls_, Apr 30 2004
%E A094057 a(23)-a(36) from _Arkadiusz Wesolowski_, Jul 20 2012
%E A094057 Name edited and a(0)=1 inserted by _Hans J. H. Tuenter_, Jul 23 2025