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

%I A350701 #19 Jan 27 2022 20:43:29
%S A350701 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,
%T A350701 154,195,248,315,402,511,649,826,1052,1337,1700,2164,2751,3501,4452,
%U A350701 5664,7204,9164,11656,14828,18861,23991,30518,38818,49379,62810,79896
%N A350701 a(n) is the number of squares strictly between Fibonacci(n) and Fibonacci(n+1).
%C A350701 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.
%H A350701 Karl-Heinz Hofmann, <a href="/A350701/b350701.txt">Table of n, a(n) for n = 0..1000</a>
%e A350701 Strictly between Fibonacci(9) = 34 and Fibonacci(10) = 55 are the 2 squares 36 and 49. So a(9) = 2.
%o A350701 (Python)
%o A350701 from math import isqrt
%o A350701 from sympy import fibonacci as fi
%o A350701 print([0,0] + [(isqrt(fi(k+1)-1) - isqrt(fi(k))) for k in range(2, 55)])
%o A350701 (Python)
%o A350701 from math import isqrt
%o A350701 from gmpy2 import fib2
%o A350701 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
%o A350701 (PARI) a(n)={if(n<=1, 0, sqrtint(fibonacci(n+1)-1) - sqrtint(fibonacci(n)))} \\ _Andrew Howroyd_, Jan 25 2022
%Y A350701 Cf. A000045, A000290, A061287, A076777.
%K A350701 nonn
%O A350701 0,10
%A A350701 _Karl-Heinz Hofmann_, Jan 24 2022