A030273 -id:A030273 - cp's OEIS frontend

A298938 Number of ordered ways of writing n^3 as a sum of n squares of nonnegative integers.

1, 1, 1, 4, 5, 686, 13942, 455988, 13617853, 454222894, 18323165948, 802161109047, 42149084452070, 2481730049781672, 157265294178424356, 10977302934685469078, 812821237985857557677, 64539935903231450294134, 5504599828399250884049308

Offset: 0

Ilya Gutkovskiy, Jan 29 2018

nonn

			a(4) = 5 because we have [64, 0, 0, 0], [16, 16, 16, 16], [0, 64, 0, 0], [0, 0, 64, 0] and [0, 0, 0, 64].

Index entries for sequences related to sums of squares

Cf. A000290, A000578, A006456, A030273, A037444, A066535, A218494, A232173, A287617, A298329, A298330, A298935, A298936, A298939.

Table[SeriesCoefficient[(1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^3}], {n, 0, 18}]

a(n) = [x^(n^3)] (Sum_{k>=0} x^(k^2))^n.

A298939 Number of ordered ways of writing n^3 as a sum of n squares of positive integers.

Original entry on oeis.org

1, 1, 1, 4, 1, 286, 7582, 202028, 6473625, 226029577, 8338249868, 391526193477, 19990594900630, 1159906506684446, 74890158861242740, 5119732406649036418, 380146984328280974281, 30198665638519565614034, 2555354508318427693497565

Offset: 0

Ilya Gutkovskiy, Jan 29 2018

nonn

			a(3) = 4 because we have [25, 1, 1], [9, 9, 9], [1, 25, 1] and [1, 1, 25].

Index entries for sequences related to sums of squares

Cf. A000290, A000578, A006456, A030273, A037444, A066535, A218494, A232173, A287617, A298329, A298330, A298935, A298937, A298938.

Table[SeriesCoefficient[(-1 + EllipticTheta[3, 0, x])^n/2^n, {x, 0, n^3}], {n, 0, 18}]

a(n) = [x^(n^3)] (Sum_{k>=1} x^(k^2))^n.

A331884 Number of compositions (ordered partitions) of n^2 into distinct squares.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 7, 1, 31, 123, 151, 121, 897, 7351, 5415, 14881, 48705, 150583, 468973, 1013163, 1432471, 1730023, 50432107, 14925241, 125269841, 74592537, 241763479, 213156871, 895153173, 7716880623, 2681163865, 3190865761, 22501985413, 116279718801

Offset: 0

Ilya Gutkovskiy, Jan 30 2020

nonn

			a(5) = 3 because we have [25], [16, 9] and [9, 16].

Cf. A000290, A006456, A030273, A032020, A037444, A105152, A224366, A232173, A280129, A298640, A331844.

b:= proc(n, i, p) option remember;
      `if`(i*(i+1)*(2*i+1)/6n, 0, b(n-i^2, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n^2, n, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 30 2020

b[n_, i_, p_] := b[n, i, p] = If[i(i+1)(2i+1)/6 < n, 0, If[n == 0, p!, If[i^2 > n, 0, b[n - i^2, i - 1, p + 1]] + b[n, i - 1, p]]];
a[n_] := b[n^2, n, 0];
a /@ Range[0, 35] (* Jean-François Alcover, Nov 08 2020, after Alois P. Heinz *)

a(n) = A331844(A000290(n)).

a(24)-a(34) from Alois P. Heinz, Jan 30 2020

A307608 Number of partitions of n^2 into consecutive positive squares.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Ilya Gutkovskiy, Apr 18 2019

nonn

			29^2 = 20^2 + 21^2, so a(29) = 2.

Index entries for sequences related to sums of squares

Cf. A000290, A030273, A034705, A037444, A097812, A151557, A296338.

a(n) = [x^(n^2)] Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(k^2).
a(n) = A296338(A000290(n)).
a(n) >= 2 for n in A097812.

A385963 a(n) is the maximum number of distinct positive integers whose sum of squares is equal to n^2.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 4, 5, 5, 5, 6, 7, 7, 7, 8, 8, 9, 9, 9, 9, 11, 10, 11, 11, 11, 11, 11, 13, 12, 12, 13, 14, 13, 14, 14, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 22, 24, 24, 24, 24, 24, 25

Offset: 0

Gonzalo Martínez, Jul 13 2025

nonn

An upper bound is a(n) <= r for the largest r with 1^2 + ... + r^2 <= n^2, and with equality only at n = 0,1,24, the latter being a(70) = 24 (see comments A001032).

			For n = 11, there are A030273(11) = 4 partitions of 11^2 into distinct squares: {11^2}, {2^2, 6^2, 9^2}, {1^2, 2^2, 4^2, 10^2}, {1^2, 2^2, 4^2, 6^2, 8^2}, where the largest cardinality of these sets is 5. Therefore, a(11) = 5.

David A. Corneth, Table of n, a(n) for n = 0..10000
David A. Corneth, n, a(n), one tuple of size a(n) where sum of elements of tuples is n^2

Cf. A030273, A001032.

a(n)=poldegree(polcoef(prod(k=1, n, 1 + y*x^(k^2), 1 + O(x^(n^2+1))), n^2)) \\ Andrew Howroyd, Jul 13 2025

More terms from Andrew Howroyd, Jul 13 2025

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A298938 Number of ordered ways of writing n^3 as a sum of n squares of nonnegative integers.

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A298939 Number of ordered ways of writing n^3 as a sum of n squares of positive integers.

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A331884 Number of compositions (ordered partitions) of n^2 into distinct squares.

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A307608 Number of partitions of n^2 into consecutive positive squares.

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A385963 a(n) is the maximum number of distinct positive integers whose sum of squares is equal to n^2.

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