A334138 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.

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, 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, 4, 4, 5, 6, 2, 4, 4, 4, 4, 2, 2, 2, 7, 10, 5, 4, 4, 5, 7, 3, 4, 6, 3

Offset: 0

Author

Zhi-Wei Sun, Apr 15 2020

Keywords

• nonn

Comments

Note that {y*(2*y+1): y is an integer} = {n*(n+1)/2: n = 0,1,...}.

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.

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.

(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.

(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.

(iv) The set {x^4+y(5y+3)+z(3z+1)/2: x,y,z are integers} contains all nonnegative integers except for 2899.

(v) The set {x^4+y(5y+4)+z(3z+1)/2: x,y,z are integers} contains all nonnegative integers except for 17960.

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

Examples

			a(9) = 1 with 9 = 1^4 + (-2)*(2*(-2)+1) + (-1)*(3*(-1)+1).
a(554) = 1 with 554 = 2^4 + 16*(2*16+1) + (-2)*(3*(-2)+1).
a(555) = 1 with 555 = 2^4 + (-5)*(2*(-5)+1) + (-13)*(3*(-13)+1).
a(25713) = 1 with 25713 = 8^4 + (-85)*(2*(-85)+1) + 49*(3*49+1).
a(80488) = 1 with 80488 = 0^4 + (-196)*(2*(-196)+1) + (-36)*(3*(-36)+1).
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
• Zhi-Wei Sun, Universal sums of three quadratic polynomials, Sci. China Math. 63 (2020), 501-520.

Crossrefs

Cf. A000217, A000583, A001082, A001318, A152749, A270566, A334147.

Programs

Mathematica

QQ[n_]:=QQ[n]=IntegerQ[Sqrt[12n+1]];
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]