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A306459 Number of ways to write n as w^3 + C(x+2,3) + C(y+2,3) + C(z+2,3), where w,x,y,z are nonnegative integers with x <= y <= z, and C(m,k) denotes the binomial coefficient m!/(k!*(m-k)!).

%I A306459 #27 Feb 21 2019 01:09:24
%S A306459 1,2,2,2,2,2,2,1,2,3,3,3,4,3,2,2,2,1,2,2,4,4,4,2,2,3,2,1,4,4,4,4,4,2,
%T A306459 1,3,4,3,4,4,4,5,3,2,3,4,2,4,5,3,2,4,2,1,1,3,4,6,4,2,3,4,2,3,5,4,5,7,
%U A306459 5,2,4,4,4,3,3,4,6,4,4,2,2,2,4,3,6,6,5,4,6,3,2,3,6,4,6,4,4,4,4,3,3
%N A306459 Number of ways to write n as w^3 + C(x+2,3) + C(y+2,3) + C(z+2,3), where w,x,y,z are nonnegative integers with x <= y <= z, and C(m,k) denotes the binomial coefficient m!/(k!*(m-k)!).
%C A306459 Conjecture: a(n) > 0 for all n >= 0. In other words, each nonnegative integer can be written as the sum of a nonnegative cube and three tetrahedral numbers.
%C A306459 It seems that a(n) = 1 only for n = 0, 7, 17, 27, 34, 53, 54, 110, 118, 163, 207, 263, 270, 309, 362, 443, 1174, 1284.
%C A306459 We have verified a(n) > 0 for all n = 0..2*10^6.
%H A306459 Zhi-Wei Sun, <a href="/A306459/b306459.txt">Table of n, a(n) for n = 0..10000</a>
%e A306459 a(0) = 1 with 0 = 0^3 + C(2,3) + C(2,3) + C(2,3).
%e A306459 a(17) = 1 with 17 = 2^3 + C(3,3) + C(4,3) + C(4,3).
%e A306459 a(27) = 1 with 27 = 3^3 + C(2,3) + C(2,3) + C(2,3).
%e A306459 a(362) = 1 with 362 = 0^3 + C(6,3) + C(8,3) + C(13,3).
%e A306459 a(443) = 1 with 443 = 3^3 + C(5,3) + C(10,3) + C(13,3).
%e A306459 a(1174) = 1 with 1174 = 1^3 + C(9,3) + C(10,3) + C(19,3).
%e A306459 a(1284) = 1 with 1284 = 10^3 + C(7,3) + C(9,3) + C(11,3).
%t A306459 f[n_]:=f[n]=Binomial[n+2,3];
%t A306459 CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
%t A306459 tab={};Do[r=0;Do[If[f[x]>n/3,Goto[cc]];Do[If[f[y]>(n-f[x])/2,Goto[bb]];Do[If[f[z]>n-f[x]-f[y],Goto[aa]];If[CQ[n-f[x]-f[y]-f[z]],r=r+1],{z,y,n-f[x]-f[y]}];Label[aa],{y,x,(n-f[x])/2}];Label[bb],{x,0,n/3}];Label[cc];tab=Append[tab,r],{n,0,100}];Print[tab]
%Y A306459 Cf. A000292, A000578, A000797, A262813, A306460, A306462, A306471, A306477.
%K A306459 nonn
%O A306459 0,2
%A A306459 _Zhi-Wei Sun_, Feb 20 2019