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

Author

Hoang Xuan Thanh, Jul 16 2025

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Programs

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 *)

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;}

Formula

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