A350701 a(n) is the number of squares strictly between Fibonacci(n) and Fibonacci(n+1).

0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 2, 2, 3, 4, 5, 7, 8, 11, 14, 18, 22, 29, 36, 46, 58, 75, 95, 120, 154, 195, 248, 315, 402, 511, 649, 826, 1052, 1337, 1700, 2164, 2751, 3501, 4452, 5664, 7204, 9164, 11656, 14828, 18861, 23991, 30518, 38818, 49379, 62810, 79896

Offset: 0

Author

Karl-Heinz Hofmann, Jan 24 2022

Keywords

• nonn

Comments

Terms a(0..3) are of course 0, because A000045(4) = 3 and A000045(5) = 5 are the first terms which are letting room for at least 1 integer.

Examples

			Strictly between Fibonacci(9) = 34 and Fibonacci(10) = 55 are the 2 squares 36 and 49. So a(9) = 2.
		

Links

• Karl-Heinz Hofmann, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000045, A000290, A061287, A076777.

Programs

PARI

a(n)={if(n<=1, 0, sqrtint(fibonacci(n+1)-1) - sqrtint(fibonacci(n)))} \\ Andrew Howroyd, Jan 25 2022

Python

from math import isqrt
from sympy import fibonacci as fi
print([0,0] + [(isqrt(fi(k+1)-1) - isqrt(fi(k))) for k in range(2, 55)])

Python

from math import isqrt
from gmpy2 import fib2
def A350701(n): return 0 if n <= 1 else (lambda x:isqrt(x[0]-1)-isqrt(x[1]))(fib2(n+1)) # Chai Wah Wu, Jan 25 2022