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%I #20 Apr 16 2020 09:36:03
%S 1,2,2,3,3,3,3,2,2,1,3,4,2,2,3,4,4,4,3,3,4,5,2,2,4,5,6,2,3,3,6,7,4,4,
%T 2,3,5,3,2,4,5,5,1,2,2,6,8,5,3,2,4,4,5,3,4,5,5,1,4,5,5,5,2,3,0,3,6,4,
%U 4,4,5,6,2,4,4,4,4,2,2,2,7,10,5,4,4,5,7,3,4,6,3
%N Number of ways to write n as x^4 + y*(2*y+1) + z*(3*z+1), where x is a nonnegative integer, and y and z are integers.
%C Note that {y*(2*y+1): y is an integer} = {n*(n+1)/2: n = 0,1,...}.
%C Conjecture 1: a(n) = 0 only for n = 64. In other words, any nonnegative integer n not equal to 64 can be written as x^4 + y*(2*y+1) + z*(3*z+1) with x,y,z integers.
%C Conjecture 2: (i) The set {x^4+y^2+z(3z+1)/2: x,y,z are integers} contains all nonnegative integers except for 455.
%C (ii) The set {x^4+y(3y+1)+z(5z+1)/2: x,y,z are integers} contains all nonnegative integers except for 59, and the set {x^4+y(3y+1)+z(5z+3)/2: x,y,z are integers} contains all nonnegative integers except for 856.
%C (iii) The set {x^4+y(3y+1)+z(3z+2): x,y,z are integers} = {x^4+3y(y+1)/2+z(3z+1)/2: x,y,z are integers} contains all nonnegative integers except for 1975.
%C (iv) The set {x^4+y(5y+3)+z(3z+1)/2: x,y,z are integers} contains all nonnegative integers except for 2899.
%C (v) The set {x^4+y(5y+4)+z(3z+1)/2: x,y,z are integers} contains all nonnegative integers except for 17960.
%C We have verified Conjecture 1 for n up to 10^8, parts (i) and (iii) of Conjecture 2 for n up to 5*10^7, and parts (ii), (iv) and (v) of Conjecture 2 for n up to 2*10^6. See also A334147 for the list of those numbers n with a(n) = 1. - _Zhi-Wei Sun_, Apr 16, 2020
%H Zhi-Wei Sun, <a href="/A334138/b334138.txt">Table of n, a(n) for n = 0..10000</a>
%H Zhi-Wei Sun, <a href="https://doi.org/10.1007/s11425-017-9354-4">Universal sums of three quadratic polynomials</a>, Sci. China Math. 63 (2020), 501-520.
%e a(9) = 1 with 9 = 1^4 + (-2)*(2*(-2)+1) + (-1)*(3*(-1)+1).
%e a(554) = 1 with 554 = 2^4 + 16*(2*16+1) + (-2)*(3*(-2)+1).
%e a(555) = 1 with 555 = 2^4 + (-5)*(2*(-5)+1) + (-13)*(3*(-13)+1).
%e a(25713) = 1 with 25713 = 8^4 + (-85)*(2*(-85)+1) + 49*(3*49+1).
%e a(80488) = 1 with 80488 = 0^4 + (-196)*(2*(-196)+1) + (-36)*(3*(-36)+1).
%t QQ[n_]:=QQ[n]=IntegerQ[Sqrt[12n+1]];
%t tab={};Do[r=0;Do[If[QQ[n-x^4-y(2y+1)],r=r+1],{x,0,n^(1/4)},{y,-Floor[(Sqrt[8(n-x^4)+1]+1)/4],(Sqrt[8(n-x^4)+1]-1)/4}];tab=Append[tab,r],{n,0,90}];Print[tab]
%Y Cf. A000217, A000583, A001082, A001318, A152749, A270566, A334147.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, Apr 15 2020
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