A216223 Distance from Fibonacci(n) to the next perfect square.

0, 0, 0, 2, 1, 4, 1, 3, 4, 2, 9, 11, 0, 23, 23, 15, 37, 3, 17, 44, 124, 79, 245, 243, 288, 51, 408, 718, 285, 1295, 1529, 1652, 267, 2306, 4434, 1979, 144, 9239, 11840, 4223, 19534, 5283, 29865, 19604, 46492, 45551, 67706, 16008, 92593, 145155, 102696, 276775

Offset: 0

Author

Carmine Suriano, Mar 13 2013

Keywords

• nonn

Comments

Difference between y^2 and Fibonacci(n), y being next integer square root of Fibonacci(n). a(n)=0 only for n = 0, 1, 2, 12.

a(n) is a square for n = 0, 1, 2, 4, 5, 6, 8, 10, 12, 36.

Examples

			a(5) = 4 since Fibonacci(5)=5 that differs 4 to next square that is 9.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A000045, A190993.

Programs

Maple

a:= n-> (f-> ceil(sqrt(f))^2-f)((<<0|1>, <1|1>>^n)[1, 2]):
seq(a(n), n=0..51);  # Alois P. Heinz, Oct 26 2022

Mathematica

Table[k = Ceiling[Sqrt[Fibonacci[n]]]; k^2 - Fibonacci[n], {n, 0, 60}] (* T. D. Noe, Mar 13 2013 *)

Formula

a(n) = floor(sqrt(Fibonacci(n))+1)^2-Fibonacci(n) if n<>1, 2, 12; else a(n)=0.