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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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified June 15 02:56 EDT 2021. Contains 345042 sequences. (Running on oeis4.)