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A306439 Number of ways to write n as x*(3x+1)/2 + y*(3y+1)/2 + z*(3z+1) + 3w*(3w+1)/2, where x,y,z,w are nonnegative integers with x <= y.

%I A306439 #9 Feb 16 2019 19:29:08
%S A306439 1,0,1,0,2,0,2,1,2,1,2,1,1,2,3,2,1,2,2,2,2,4,2,3,2,3,2,3,4,3,4,1,5,1,
%T A306439 5,3,5,4,3,4,5,1,5,3,4,4,3,7,2,4,4,7,6,6,4,4,5,3,7,5,5,8,6,7,3,6,8,6,
%U A306439 5,4,3,4,6,7,3,7,6,10,7,5,9,3,11,4,9,7,7,10,5,9,7,7,10,8,7,5,5,9,5,9,9
%N A306439 Number of ways to write n as x*(3x+1)/2 + y*(3y+1)/2 + z*(3z+1) + 3w*(3w+1)/2, where x,y,z,w are nonnegative integers with x <= y.
%C A306439 Conjecture 1: a(n) > 0 for all n > 5, and a(n) = 1 only for n = 0, 2, 7, 9, 11, 12, 16, 31, 33, 41.
%C A306439 Conjecture 2: Let n be any integer greater than 9, and let p(x) denote x*(3x+1)/2. For each c = 2, 4, 9, we can write n as p(x) + 2*p(y) + 3*p(z) + c*p(w) with x,y,z,w nonnegative integers.
%C A306439 See also Conjecture 5.2 of the linked 2016 paper.
%H A306439 Zhi-Wei Sun, <a href="/A306439/b306439.txt">Table of n, a(n) for n = 0..10000</a>
%H A306439 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2015.10.014">A result similar to Lagrange's theorem</a>, J. Number Theory 162(2016), 190-211.
%e A306439 a(12) = 1 with 12 = 0*(3*0+1)/2 + 1*(3*1+1)/2 + 1*(3*1+1) + 3*1*(3*1+1)/2.
%e A306439 a(31) = 1 with 31 = 1*(3*1+1)/2 + 3*(3*3+1)/2 + 2*(3*2+1) + 3*0*(3*0+1)/2.
%e A306439 a(33) = 1 with 33 = 2*(3*2+1)/2 + 4*(3*4+1)/2 + 0*(3*0+1) + 3*0*(3*0+1)/2.
%e A306439 a(41) = 1 with 41 = 3*(3*3+1)/2 + 4*(3*4+1)/2 + 0*(3*0+1) + 3*0*(3*0+1)/2.
%t A306439 PQ[n_]:=PQ[n]=IntegerQ[Sqrt[24n+1]]&&Mod[Sqrt[24n+1],6]==1;
%t A306439 tab={};Do[r=0;Do[If[PQ[n-3x(3x+1)/2-y(3y+1)-z(3z+1)/2],r=r+1],{x,0,(Sqrt[8n+1]-1)/6},{y,0,(Sqrt[12(n-3x(3x+1)/2)+1]-1)/6},{z,0,(Sqrt[12(n-3x(3x+1)/2-y(3y+1))+1]-1)/6}];tab=Append[tab,r],{n,0,100}];Print[tab]
%Y A306439 Cf. A005449, A306382, A306383.
%K A306439 nonn
%O A306439 0,5
%A A306439 _Zhi-Wei Sun_, Feb 15 2019