A362409 a(n) is the least number that is the sum of a Fibonacci number and a square in exactly n ways.

15, 7, 3, 1, 17

Offset: 0

Author

Robert Israel, Apr 21 2023

Keywords

• nonn
• more

Comments

a(n) is the least k such that there are n pairs (i,j) of nonnegative integers such that A000045(i) + j^2 = k.

We count A000045(1) and A000045(2) separately, even though both are 1.

a(5) > 10^20, if it exists. - Martin Ehrenstein, May 01 2023

Examples

			a(0) = 15 because 15 is not the sum of a Fibonacci number and a square.
a(1) = 7 because 7 = A000045(4) + 2^2 is the sum of a Fibonacci number and a square in one way.
a(2) = 3 because 3 = A000045(3) + 1^2 = A000045(4) + 0^2.
a(3) = 1 because 1 = A000045(0) + 1^2 = A000045(1) + 0^2 = A000045(2) + 0^2.
a(4) = 17 because 17 = A000045(1) + 4^2 = A000045(2) + 4^2 = A000045(6) + 3^2 = A000045(7) + 2^2.
		

Crossrefs

Cf. A000045, A362434, A362503.

Programs

Maple

N:= 10^8: # 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:
W:= Array(0..max(V)):
for i from 1 to N do
  w:= V[i];
  if W[w] = 0 then W[w]:= i fi
od:
convert(W,list);