A386235 Number of partitions (p, q, r) of n into positive integers such that p + 11*q + 13*r is a perfect square.

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 6, 3, 7, 8, 6, 7, 6, 9, 8, 9, 9, 9, 15, 11, 13, 16, 13, 17, 15, 16, 17, 16, 19, 19, 23, 19, 21, 27, 23, 26, 24, 25, 29, 27, 28, 30, 32, 32, 34, 37, 35, 36, 37, 38, 40, 38, 38, 44, 46, 43, 46, 48, 50, 50, 48, 50, 50, 54, 52, 56, 60, 54, 64, 63, 62, 64

Offset

3,8

Comments

For n >= 3 then there exists (p, q, r) | p + q + r = n such that p + 11*q + 13*r is a perfect square. This has been proven by Sylvester's theorem.

Links

Hoang Xuan Thanh, Table of n, a(n) for n = 3..20000

Formula

Conjecture: a(n) ~ K * n^(3/2) where K = 0.0914... from a(10000) = 91413 and a(20000) = 258667.

K = (1 - 66*sqrt(11) + 65*sqrt(13))/180. - Hoang Xuan Thanh, Sep 13 2025

Example

n = 12: (4,4,4); 4 + 11*4 + 13*4 = 10^2; (7,4,1); 7 + 11*4 + 13*1 = 8^2; so a(12) = 2.

Mathematica

a[n_]:=Module[{cnt=0, p, m2}, Do[Do[p=n-q-r; m2=p +11*q+13*r; If[IntegerQ[Sqrt[m2]], cnt++], {r, 1, n - q - 1}], {q, 1, n-2}]; cnt]; Array[a, 78, 3] (* James C. McMahon, Jul 22 2025 *)

Prog

(PARI) a(n) = {my(cnt = 0); for (q = 1, n-2, for (r = 1, n - q - 1, p = n - q - r; m2 = p + 11*q + 13*r; if (issquare(m2), cnt++); ); ); cnt; }

Crossrefs

Sequence in context: A216393 A045812 A391211 * A064877 A394207 A008363

Adjacent sequences: A386232 A386233 A386234 * A386236 A386237 A386238

Keyword

nonn

Author

Hoang Xuan Thanh, Jul 16 2025

Status

approved