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A316491
Number of ways to represent 8*n + 4 as the sum of four distinct odd squares.
1
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
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
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Jul 29 2018
STATUS
approved