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A385248 Number of digits in the decimal expansion of Fibonacci(2^n).

%I A385248 #98 Aug 14 2025 02:57:18
%S A385248 1,1,1,2,3,7,14,27,54,107,214,428,856,1712,3424,6848,13696,27393,
%T A385248 54785,109570,219140,438279,876558,1753116,3506231,7012462,14024923,
%U A385248 28049846,56099693,112199385,224398770,448797540,897595080,1795190160,3590380321,7180760641
%N A385248 Number of digits in the decimal expansion of Fibonacci(2^n).
%C A385248 Binet's formula is Fibonacci(k) = (phi^k - psi^k)/sqrt(5), with phi being the golden ratio (1 + sqrt(5))/2, and psi = (1 - sqrt(5))/2. For even values of k, Fibonacci(k) = floor((phi^k)/sqrt(5)) since psi^(2*k)/sqrt(5) < 0.17^k for all k > 0, and from which the formula below.
%H A385248 Paolo Xausa, <a href="/A385248/b385248.txt">Table of n, a(n) for n = 0..3000</a>
%F A385248 a(n) = A055642(A058635(n)).
%F A385248 a(n) = A060384(A000079(n)).
%F A385248 a(n) = floor(2^n * log10(phi) - (1/2) * log10(5)) + 1, for n >= 1.
%F A385248 Limit_{n->oo} a(n+1)/a(n) = 2.
%p A385248 a:= n-> `if`(n=0, 1, floor(2^n*log[10]((1+sqrt(5))/2)-log[10](5)/2)+1):
%p A385248 seq(a(n), n=0..35);  # _Alois P. Heinz_, Jul 30 2025
%t A385248 a[n_] := IntegerLength[Fibonacci[2^n]]; Array[a, 30, 0] (* _Amiram Eldar_, Jul 30 2025 *)
%t A385248 A385248[n_] := If[n == 0, 1, Floor[2^n*Log10[GoldenRatio] - Log10[5]/2] + 1];
%t A385248 Array[A385248, 50, 0] (* _Paolo Xausa_, Aug 07 2025 *)
%o A385248 (Python)
%o A385248 from sympy import Rational, log, sqrt # uses symbolic computation
%o A385248 phi = (1+sqrt(5))/2
%o A385248 def a(n): return 1 if n==0 or n==1 else int(2**n *log(phi)/log(10)-Rational(1,2)*log(5)/log(10))+1
%o A385248 (PARI) a(n) = #Str(fibonacci(2^n)); \\ _Michel Marcus_, Jul 30 2025
%Y A385248 Cf. A000079, A000045, A001622, A055642, A058635, A060384, A068070.
%K A385248 nonn,base
%O A385248 0,4
%A A385248 _Juande Santander-Vela_, Jul 28 2025
%E A385248 More terms from _Michel Marcus_, Jul 30 2025
%E A385248 a(29)-a(35) from _Amiram Eldar_, Jul 30 2025