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
Keywords
Examples
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.
Links
- Robert Israel, Table of n, a(n) for n = 0..10000
Programs
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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 *)
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