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

0,2

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

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.

Example

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).

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]

Crossrefs

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

Sequence in context: A215406 A231717 A253315 * A210480 A321859 A321860

Adjacent sequences: A334135 A334136 A334137 * A334139 A334140 A334141

Keyword

nonn

Author

Zhi-Wei Sun, Apr 15 2020

Status

approved