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A306790 Number of ways to write n as x*(3*x+1)/2 + y*(y+1)*(y+2)/2 + z*(z+1)*(z+2)/6, where x is a nonzero integer, and y and z are nonnegative integers.

%I A306790 #10 Mar 11 2019 12:39:30
%S A306790 1,2,1,1,4,3,1,3,3,1,3,3,2,4,5,3,3,4,4,2,2,4,3,3,6,3,4,4,3,3,2,4,3,3,
%T A306790 7,5,4,5,7,3,3,6,3,4,5,3,5,4,3,3,3,6,4,3,5,3,6,5,1,5,7,6,4,5,4,6,5,2,
%U A306790 5,7,7,5,7,5,5,3,5,3,2,6,5,7,5,1,6,5,7,3,5,6,6,6,4,3,4,7,4,3,6,6
%N A306790 Number of ways to write n as x*(3*x+1)/2 + y*(y+1)*(y+2)/2 + z*(z+1)*(z+2)/6, where x is a nonzero integer, and y and z are nonnegative integers.
%C A306790 Conjecture: a(n) > 0 for all n > 0. In other words, each positive integer n can be written as the sum of a positive generalized pentagonal number, a tetrahedral number and a tetrahedral number times three.
%C A306790 This has been verified for all n = 1..2*10^7.
%H A306790 Zhi-Wei Sun, <a href="/A306790/b306790.txt">Table of n, a(n) for n = 1..10000</a>
%e A306790 a(59) = 1 with 59 = (-3)*(3*(-3)+1)/2 + 2*3*4/2 + 5*6*7/6.
%e A306790 a(19694) = 1 with 19694 = 20*(3*20+1)/2 + 10*11*12/2 + 47*48*49/6.
%e A306790 a(19919) = 1 with 19919 = (-45)*(3*(-45)+1)/2 + 30*31*32/2 + 22*23*24/6.
%e A306790 a(33989) = 1 with 33989 = 55*(3*55+1)/2 + 20*21*22/2 + 52*53*54/6.
%e A306790 a(60769) = 1 with 60769 = 46*(3*46+1)/2 + 47*48*49/2 + 23*24*25/6.
%t A306790 f[n_]:=f[n]=Binomial[n+2,3]; PQ[n_]:=PQ[n]=IntegerQ[Sqrt[24n+1]];
%t A306790 tab={};Do[r=0;Do[If[f[x]>=n/3,Goto[cc]];Do[If[f[y]>=n-3*f[x],Goto[bb]];If[PQ[n-3*f[x]-f[y]],r=r+1];Label[aa],{y,0,n-1-3*f[x]}];Label[bb],{x,0,(n-1)/3}];Label[cc];tab=Append[tab,r],{n,1,100}];Print[tab]
%Y A306790 Cf. A000292, A001318, A306460, A306477.
%K A306790 nonn
%O A306790 1,2
%A A306790 _Zhi-Wei Sun_, Mar 10 2019