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A338696 Number of ways to write n as x^3 + y^2 + z*(3*z+2), where x and y are nonnegative integers, and z is an integer.

%I A338696 #17 May 07 2021 01:11:01
%S A338696 3,3,1,1,3,3,1,2,6,5,1,2,3,2,1,3,8,4,0,2,3,4,1,3,7,4,2,3,3,3,3,4,7,4,
%T A338696 2,4,5,5,1,2,7,5,3,6,5,1,2,3,7,5,2,6,2,2,1,2,10,5,2,4,2,1,1,7,11,8,2,
%U A338696 5,6,5,3,4,11,3,1,5,5,2,1,5,8,6,4,5,5,5,3,2,9,7,2,6,4,5,1,5,10,5,2,4
%N A338696 Number of ways to write n as x^3 + y^2 + z*(3*z+2), where x and y are nonnegative integers, and z is an integer.
%C A338696 Conjecture: a(n) > 0 except for n = 19.
%C A338696 We have verified this for n up to 5*10^6.
%C A338696 As z*(3*z+2) = floor((3*z+1)^2/3) and 19 = 0^3 + 4^2 + floor(3^2/3), the conjecture implies that each n = 0,1,... can be written as x^3 + y^2 + floor(z^2/3) with x,y,z nonnegative integers.
%H A338696 Zhi-Wei Sun, <a href="/A338696/b338696.txt">Table of n, a(n) for n = 1..10000</a>
%e A338696 a(63) = 1 with 63 = 3^3 + 6^2 + 0*(3*0+2).
%e A338696 a(327) = 1 with 327 = 5^3 + 13^2 + 3*(3*3+2).
%e A338696 a(478) = 1 with 478 = 6^3 + 1^2 + 9*(3*9+2).
%e A338696 a(847) = 1 with 847 = 1^3 + 29^2 + 1*(3*1+2).
%e A338696 a(1043) = 1 with 1043 = 3^3 + 20^2 + 14*(3*14+2).
%e A338696 a(3175) = 1 with 3175 = 5^3 + 35^2 + (-25)*(3*(-25)+2).
%t A338696 OctQ[n_]:=OctQ[n]=IntegerQ[Sqrt[3n+1]];
%t A338696 tab={};Do[r=0;Do[If[OctQ[n-x^3-y^2],r=r+1],{x,0,n^(1/3)},{y,0,Sqrt[n-x^3]}];tab=Append[tab,r],{n,1,100}];Print[tab]
%Y A338696 Cf. A000290, A000578, A001082, A262813, A270469, A338686, A338687.
%K A338696 nonn
%O A338696 1,1
%A A338696 _Zhi-Wei Sun_, Apr 24 2021