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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-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

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