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%I #10 Jul 20 2023 14:49:35
%S 1,2,3,4,4,3,3,2,3,5,5,6,6,3,4,1,4,6,7,10,7,5,4,2,5,8,8,9,9,6,6,2,6,
%T 10,8,13,9,6,7,5,5,8,6,9,10,6,9,4,5,9,6,13,10,7,11,6,8,10,8,10,12,9,9,
%U 7,8,13,10,16,12,6,12,8,10,13,12,13,12,8,11,7,10,16,15,17,16,6,11,7,12,16,11,16,9,10,5,6,10,15,17,18,16
%N Number of ordered ways to write n as w^3 + 2*x^3 + y^2 + 2*z^2, where w, x, y and z are nonnegative integers.
%C Conjecture: We have {a*w^3+b*x^3+c*y^2+d*z^2: w,x,y,z = 0,1,2,...} = {0,1,2,...} if (a,b,c,d) is among the following 63 quadruples:
%C (1,1,1,2),(1,1,2,4),(1,2,1,1),(1,2,1,2),(1,2,1,3),(1,2,1,4),(1,2,1,6),(1,2,1,13),(1,2,2,3),(1,2,2,4),(1,2,2,5),(1,3,1,1),(1,3,1,2),(1,3,1,3),(1,3,1,5),(1,3,1,6),(1,3,2,3),(1,3,2,4),(1,3,2,5),(1,4,1,1),(1,4,1,2),(1,4,1,3),(1,4,2,2),(1,4,2,3),(1,4,2,5),(1,5,1,1),(1,5,1,2),(1,6,1,1),(1,6,1,3),(1,7,1,2),(1,8,1,2),(1,9,1,2),(1,9,2,4),(1,10,1,2),(1,11,1,2),(1,11,2,4),(1,12,1,2),(1,14,1,2),(1,15,1,2),(2,3,1,1),(2,3,1,2),(2,3,1,3),(2,3,1,4),(2,4,1,1),(2,4,1,2),(2,4,1,6),(2,4,1,8),(2,4,1,10),(2,5,1,3),(2,6,1,1),(2,7,1,3),(2,8,1,1),(2,8,1,4),(2,10,1,1),(2,13,1,1),(3,4,1,2),(3,5,1,2),(3,7,1,2),(3,9,1,2),(4,5,1,2),(4,6,1,2),(4,8,1,2),(4,11,1,2).
%C Conjecture verified up to 10^11 for all quadruples. - _Mauro Fiorentini_, Jul 18 2023
%H Zhi-Wei Sun, <a href="/A262857/b262857.txt">Table of n, a(n) for n = 0..10000</a>
%e a(7) = 2 since 7 = 1^3 + 2*0^3 + 2^2 + 2*1^2 = 1^3 + 2*1^3 + 2^2 + 2*0^2.
%e a(15) = 1 since 15 = 1^3 + 2*1^3 + 2^2 + 2*2^2.
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t Do[r=0;Do[If[SQ[n-x^3-2y^3-2z^2],r=r+1],{x,0,n^(1/3)},{y,0,((n-x^3)/2)^(1/3)},{z,0,Sqrt[(n-x^3-2y^3)/2]}];Print[n," ",r];Continue,{n,0,100}]
%Y Cf. A000290, A000578, A262813, A262815, A262816, A262824, A262827.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, Oct 03 2015
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