This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A386235 #24 Jul 23 2025 09:12:16
%S A386235 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,
%T A386235 16,17,16,19,19,23,19,21,27,23,26,24,25,29,27,28,30,32,32,34,37,35,36,
%U A386235 37,38,40,38,38,44,46,43,46,48,50,50,48,50,50,54,52,56,60,54,64,63,62,64
%N A386235 Number of partitions (p, q, r) of n into positive integers such that p + 11*q + 13*r is a perfect square.
%C A386235 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.
%H A386235 Hoang Xuan Thanh, <a href="/A386235/b386235.txt">Table of n, a(n) for n = 3..20000</a>
%F A386235 Conjecture: a(n) ~ K * n^(3/2) where K = 0.0914... from a(10000) = 91413 and a(20000) = 258667.
%e A386235 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.
%t A386235 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 *)
%o A386235 (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;}
%K A386235 nonn
%O A386235 3,8
%A A386235 _Hoang Xuan Thanh_, Jul 16 2025