A362503 a(n) is the number of k such that n - A000045(k) is a square.

1, 3, 3, 2, 2, 3, 2, 1, 1, 3, 2, 1, 2, 1, 2, 0, 1, 4, 1, 1, 0, 2, 2, 0, 1, 2, 2, 1, 1, 1, 2, 0, 0, 1, 1, 1, 1, 3, 3, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 2, 3, 1, 1, 0, 1, 1, 1, 2, 0, 2, 0, 0, 1, 0, 2, 2, 1, 1, 0, 1, 2, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 2, 2, 1, 1, 1, 0, 0, 2, 1, 1, 0, 1, 1, 0, 0, 0, 2

Offset: 0

Author

Robert Israel, Apr 22 2023

Keywords

• nonn

Comments

Number of ways to write n as the sum of a Fibonacci number and a square, where A000045(1) and A000045(2) are counted as separate.

Examples

			a(5) = 3 because 5 = A000045(1) + 2^2 = A000045(2) + 2^2 = A000045(5) + 0^2.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A108852, A362409, A362434.

Programs

Maple

N:= 100: # to get terms <= N
V:= Array(0..N):
for i from 0 do
  f:= combinat:-fibonacci(i);
  if f >= N then break fi;
  s:= floor(sqrt(N-f));
  J:=[seq(f+i^2, i=0..s)];
  V[J]:= V[J] +~ 1;
od:
convert(V,list);

PARI

f(n) = my(k=1); while (fibonacci(k) <= n, k++); k; \\ A108852
a(n) = sum(k=0, f(n), issquare(n-fibonacci(k))); \\ Michel Marcus, Apr 23 2023

Formula

a(1 + A000045(6*k)^2/4) >= 4.