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A343326 Number of ways to write n as the integral part of (a^3+b^3)/2 + (c^3+d^3)/6, where a,b,c,d are nonnegative integers with a >= max{b,1} and c >= max{d,1}.

%I A343326 #13 Apr 12 2021 13:01:58
%S A343326 2,3,3,2,4,7,4,1,4,6,3,4,3,6,5,6,5,3,7,5,2,4,6,4,5,7,5,2,6,7,1,2,8,4,
%T A343326 6,5,9,10,7,4,6,7,6,2,5,8,4,6,5,5,6,4,2,7,7,2,3,9,5,3,4,6,5,7,9,7,8,8,
%U A343326 12,5,5,6,9,10,7,5,7,7,5,4,3,6,4,5,6,8,9,7,5,10,5,5,3,7,10,3,3,8,5,10,9
%N A343326 Number of ways to write n as the integral part of (a^3+b^3)/2 + (c^3+d^3)/6, where a,b,c,d are nonnegative integers with a >= max{b,1} and c >= max{d,1}.
%C A343326 Conjecture: a(n) > 0 for any nonnegative integer n.
%C A343326 This has been verified for all n = 0..10^5.
%H A343326 Zhi-Wei Sun, <a href="/A343326/b343326.txt">Table of n, a(n) for n = 0..10000</a>
%H A343326 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1504.01608">Natural numbers represented by floor(x^2/a) + floor(y^2/b) + floor(z^2/c)</a>, arXiv:1504.01608 [math.NT], 2015.
%e A343326 a(0) = 2 with 0 = floor((1^3+0^3)/2 + (1^3+0^3)/6) = floor((1^3+0^3)/2 + (1^3+1^3)/6).
%e A343326 a(7) = 1 with 7 = floor((3^3+1^3)/2 + (2^3+2^3)/6).
%e A343326 a(30) = 1 with 30 = floor((2^3+2^3)/2 + (5^3+2^3)/6).
%e A343326 a(111) = 1 with 111 = floor((6^3+1^3)/2 + (2^3+2^3)/6).
%e A343326 a(163) = 1 with 163 = floor((6^3+3^3)/2 + (5^3+5^3)/6).
%e A343326 a(219) = 1 with 219 = floor((4^3+0^3)/2 + (10^3+5^3)/6).
%t A343326 CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
%t A343326 tab={};Do[r=0;Do[If[CQ[6n+s-3(x^3+y^3)-z^3],r=r+1],{s,Boole[n==0],5},{x,1,((6n+s-1)/3)^(1/3)},{y,0,Min[x,((6n+s-1)/3-x^3)^(1/3)]},{z,0,((6n+s-3(x^3+y^3))/2)^(1/3)}];tab=Append[tab,r],{n,0,100}];Print[tab]
%Y A343326 Cf. A000578, A004999, A343368.
%K A343326 nonn
%O A343326 0,1
%A A343326 _Zhi-Wei Sun_, Apr 11 2021