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

Author

Zhi-Wei Sun, Mar 10 2022

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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]