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A306250 Number of ways to write n as x*(3x+1) + y*(3y-1) + z*(3z+2) + w*(3w-2), where x,y,z,w are nonnegative integers with x*y*z = 0.

%I A306250 #10 Feb 01 2019 01:24:29
%S A306250 1,1,1,1,1,2,2,2,2,1,3,1,1,1,3,4,3,3,2,2,2,2,2,2,4,3,3,3,3,2,4,3,3,2,
%T A306250 2,4,4,4,4,2,5,4,1,3,3,5,3,4,4,4,3,3,2,2,6,4,6,4,6,4,4,4,3,2,5,4,4,3,
%U A306250 5,4,7,4,2,2,4,8,3,4,6,4,5,6,3,5,5,6,6,5,4,5,3,4,2,4,5,6,6,7,6,1,8
%N A306250 Number of ways to write n as x*(3x+1) + y*(3y-1) + z*(3z+2) + w*(3w-2), where x,y,z,w are nonnegative integers with x*y*z = 0.
%C A306250 Conjecture: a(n) > 0 for any nonnegative integer n.
%C A306250 Clearly, a(n) <= A306242(n). We have verified a(n) > 0 for all n = 0..10^6.
%H A306250 Zhi-Wei Sun, <a href="/A306250/b306250.txt">Table of n, a(n) for n = 0..10000</a>
%H A306250 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.
%H A306250 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.07.024">On x(ax+1)+y(by+1)+z(cz+1) and x(ax+b)+y(ay+c)+z(az+d)</a>, J. Number Theory 171(2017), 275-283.
%e A306250 a(12) = 1 with 12 = 1*(3*1+1) + 0*(3*0-1) + 0*(3*0+2) + 2*(3*2-2).
%e A306250 a(42) = 1 with 42 = 0*(3*0+1) + 1*(3*1-1) + 0*(3*0+2) + 4*(3*4-2).
%e A306250 a(62) = 3 with 62 = 3*(3*3+1) + 3*(3*3-1) + 0*(3*0+2) + 2*(3*2-2)
%e A306250 = 4*(3*4+1) + 2*(3*2-1) + 0*(3*0+2) + 0*(3*0-2) = 4*(3*4+1) + 1*(3*1-1) + 0*(3*0+2) + 2*(3*2-2).
%e A306250 a(99) = 1 with 99 = 2*(3*2+1) + 0*(3*0-1) + 5*(3*5+2) + 0*(3*0-2).
%e A306250 a(118) = 1 with 118 = 0*(3*0+1) + 6*(3*6-1) + 2*(3*2+2) + 0*(3*0-2).
%t A306250 OctQ[n_]:=OctQ[n]=IntegerQ[Sqrt[3n+1]]&&(n==0||Mod[Sqrt[3n+1]+1,3]==0);
%t A306250 tab={};Do[r=0;Do[If[OctQ[n-x(3x+2)-y(3y+1)-z(3z-1)],r=r+1],{x,0,(Sqrt[3n+1]-1)/3},{y,0,(Sqrt[12(n-x(3x+2))+1]-1)/6},{z,0,If[x>0&&y>0,0,(Sqrt[12(n-x(3x+2)-y(3y+1))+1]+1)/6]}];tab=Append[tab,r],{n,0,100}];Print[tab]
%Y A306250 Cf. A000567, A045944, A049450, A049451, A255350, A306242.
%K A306250 nonn
%O A306250 0,6
%A A306250 _Zhi-Wei Sun_, Feb 01 2019