A369715 Number of digits of phi (the golden ratio) correctly approximated by Fibonacci(n+1) / Fibonacci(n).

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, 13, 13, 14, 13, 14, 15, 16, 15, 16, 17, 17, 17, 18, 18, 18, 19, 19, 20, 19, 21, 21, 22, 22, 22, 23, 23, 24, 24, 25, 24, 25, 26, 26, 27, 27, 28, 28, 28, 29, 29, 30, 30, 30

Offset: 1

Author

David Consiglio, Jr., Jan 31 2024

Keywords

• nonn
• base

Examples

			For n=1, 1/1 = 1 matches the first digit of phi (1.618033), so a(1) = 1
For n=2, 2/1 = 2 which matches no digits of phi (1.618033), so a(2) = 0
For n=12,
  F(13)/F(12) = 1.6180 55... = 233/144
  phi         = 1.6180 33...
                ^ ^^^^    a(12) = 5 matching digits
		

Links

• David Consiglio, Jr., Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000045, A001622, A048433, A048434.

Programs

Python

from math import isqrt
fib = [1,1]
terms = []
while len(terms) < 1000:
    deg = 0
    target = 0
    test = 0
    while target == test:
        target = (10**deg+isqrt(5*10**(2*deg)))//2
        test = (10**deg*(fib[-1]))//fib[-2]
        deg += 1
    terms.append(deg-1)
    fib.append(fib[-1]+fib[-2])
print(terms)