A352286 Number of ways to write n as w + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w is 0 or 1, and x,y,z are nonnegative integers.

1, 2, 2, 3, 4, 3, 2, 3, 3, 3, 2, 3, 6, 4, 3, 2, 2, 5, 5, 5, 4, 3, 4, 2, 1, 5, 5, 4, 6, 5, 3, 3, 4, 5, 4, 5, 7, 5, 4, 5, 4, 3, 3, 3, 4, 3, 3, 5, 6, 7, 6, 5, 7, 6, 4, 4, 4, 7, 5, 4, 4, 3, 7, 5, 4, 5, 5, 8, 6, 2, 2, 2, 4, 6, 4, 6, 10, 11, 6, 2, 5, 7, 7, 7, 8, 5, 3, 3, 2, 4, 4, 7, 7, 4, 6, 6, 4, 7, 8, 7, 7

Offset

0,2

Comments

Conjecture 1: a(n) = 0 only for n = 106, 744, 5469, 331269. Thus, for any nonnegative integer n not among 106, 744, 5469 and 331269, either n or n - 1 can be written as x^2 + 2*y^2 + 3*z^2 + x*y*z with x,y,z nonnegative integers.

Conjecture 2: a(n) = 1 only for n = 0, 24, 346, 360, 664, 667, 1725, 2589, 3111, 4906, 5035, 8043, 8709, 16810, 18699, 34539, 39256, 51621, 59019, 62799, 108645, 136167, 562696.

We have verified Conjectures 1 and 2 for n = 0..10^6.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Example

a(24) = 1 with 24 = 0 + 4^2 + 2*2^2 + 3*0^2 + 4*2*0.
a(346) = 1 with 346 = 1 + 15^2 + 2*3^2 + 3*2^2 + 15*3*2.
a(360) = 1 with 360 = 1 + 9^2 + 2*5^2 + 3*4^2 + 9*5*4.
a(62799) = 1 with 62799 = 1 + 16^2 + 2*169^2 + 3*2^2 + 16*169*2.
a(108645) = 1 with 108645 = 0 + 95^2 + 2*163^2 + 3*3^2 + 95*163*3.
a(136167) = 1 with 136167 = 0 + 2^2 + 2*17^2 + 3*207^2 + 2*17*207.
a(562696) = 1 with 562696 = 0 + 539^2 + 2*20^2 + 3*25^2 + 539*20*25.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[4(n-w-2y^2-3z^2)+y^2*z^2], r=r+1], {w, 0, Min[1, n]}, {z, 0, Sqrt[(n-w)/3]}, {y, 0, Sqrt[(n-w-3z^2)/2]}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]

Crossrefs

Cf. A000290, A351617, A351723, A351902, A352259.

Sequence in context: A207632 A175503 A333389 * A352259 A245909 A077769

Adjacent sequences: A352283 A352284 A352285 * A352287 A352288 A352289

Keyword

nonn

Author

Zhi-Wei Sun, Mar 10 2022

Status

approved