%I #46 Feb 03 2024 10:12:37
%S 1,0,1,2,2,2,3,3,3,4,3,5,5,6,6,6,7,6,8,8,9,8,10,10,10,11,11,11,12,11,
%T 13,13,14,13,14,15,16,15,16,17,17,17,18,18,18,19,19,20,19,21,21,22,22,
%U 22,23,23,24,24,25,24,25,26,26,27,27,28,28,28,29,29,30,30,30
%N Number of digits of phi (the golden ratio) correctly approximated by Fibonacci(n+1) / Fibonacci(n).
%H David Consiglio, Jr., <a href="/A369715/b369715.txt">Table of n, a(n) for n = 1..1000</a>
%e For n=1, 1/1 = 1 matches the first digit of phi (1.618033), so a(1) = 1
%e For n=2, 2/1 = 2 which matches no digits of phi (1.618033), so a(2) = 0
%e For n=12,
%e F(13)/F(12) = 1.6180 55... = 233/144
%e phi = 1.6180 33...
%e ^ ^^^^ a(12) = 5 matching digits
%o (Python)
%o from math import isqrt
%o fib = [1,1]
%o terms = []
%o while len(terms) < 1000:
%o deg = 0
%o target = 0
%o test = 0
%o while target == test:
%o target = (10**deg+isqrt(5*10**(2*deg)))//2
%o test = (10**deg*(fib[-1]))//fib[-2]
%o deg += 1
%o terms.append(deg-1)
%o fib.append(fib[-1]+fib[-2])
%o print(terms)
%Y Cf. A000045, A001622, A048433, A048434.
%K nonn,base
%O 1,4
%A _David Consiglio, Jr._, Jan 31 2024