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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 19 00:37 EDT 2021. Contains 345125 sequences. (Running on oeis4.)