A316491 Number of ways to represent 8*n + 4 as the sum of four distinct odd squares.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 1, 0, 1, 2, 1, 0, 2, 1, 1, 2, 3, 0, 2, 2, 0, 3, 2, 2, 3, 1, 2, 2, 2, 3, 3, 4, 0, 4, 3, 0, 6, 3, 3, 4, 3, 1, 4, 4, 3, 4, 4, 2, 6, 4, 3, 6, 3, 3, 6, 4, 3, 7, 5, 4, 5, 6, 1, 6, 6, 2, 10, 4, 5, 7, 5

Offset

0,20

Comments

Every odd square is a number of the form 8*k + 1, so every sum of four odd squares is a number of the form 8*k + 4.

A316489 lists all positive numbers of the form 8*k + 4 that cannot be expressed as the sum of four distinct odd squares; for each such number, a(k)=0.

A316834 lists all numbers that can be expressed in only one way as the sum of four distinct odd squares; each such number is of the form 8*k + 4, and for each such number, a(k)=1.

Links

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

Example

n=1: 8*1 + 4 = 12 cannot be expressed as the sum of four distinct odd squares, so a(1)=0.
n=10: 8*10 + 4 = 84 can be expressed as the sum of four distinct odd squares in only 1 way (84 = 1^2 + 3^2 + 5^2 + 7^2), so a(10)=1.
n=19: 8*19 + 4 = 156 can be expressed as the sum of four distinct odd squares in exactly 2 ways (156 = 1^2 + 3^2 + 5^2 + 11^2 = 1^2 + 5^2 + 7^2 + 9^2), so a(19)=2.

Maple

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(min(i, t)<1, 0, b(n, i-2, t)+
      `if`(i^2>n, 0, b(n-i^2, i-2, t-1))))
    end:
a:= n-> (m-> b(m, (r-> r+1-irem(r, 2))(isqrt(m)), 4))(8*n+4):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 05 2018

Mathematica

a[n_] := Count[ IntegerPartitions[8 n + 4, {4}, Range[1, Sqrt[8 n + 4], 2]^2], w_ /; Max@Differences@w < 0]; Array[a, 87, 0] (* Giovanni Resta, Aug 12 2018 *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0],
     If[Min[i, t] < 1, 0, b[n, i-2, t] +
     If[i^2 > n, 0, b[n-i^2, i-2, t-1]]]];
a[n_] := Function[m, b[m, Function[r, r+1-Mod[r, 2]][Floor@Sqrt[m]], 4]][8n+4];
a /@ Range[0, 100] (* Jean-François Alcover, May 30 2021, after Alois P. Heinz *)

Crossrefs

Cf. A316834, A316489, A316490.

Sequence in context: A265859 A271420 A099313 * A097468 A339975 A283144

Adjacent sequences: A316488 A316489 A316490 * A316492 A316493 A316494

Keyword

nonn

Author

Jon E. Schoenfield, Jul 29 2018

Status

approved