A352356 Number of ways to write 12*n + 5 as 2*x^2 + 5*y^2 + 9*z^2 + x*y*z, where x, y and z are nonnegative integers.

1, 2, 1, 3, 3, 1, 2, 4, 3, 3, 2, 5, 1, 3, 4, 3, 3, 7, 4, 2, 5, 3, 4, 3, 5, 5, 5, 6, 4, 4, 4, 2, 4, 5, 6, 3, 6, 5, 6, 5, 4, 5, 6, 7, 4, 4, 6, 4, 7, 6, 5, 3, 3, 8, 3, 7, 7, 4, 5, 7, 5, 6, 6, 8, 4, 1, 4, 7, 4, 8, 6, 5, 8, 9, 8, 4, 8, 3, 7, 4, 4, 12, 3, 4, 11, 8, 1, 6, 7, 5, 5, 8, 9, 5, 8, 12, 5, 6, 6, 6, 6

Offset: 0

Author

Zhi-Wei Sun, Mar 13 2022

Keywords

• nonn

Comments

Conjecture: For each n = 0,1,2,... we can write 12*n + 5 as 2*x^2 + 5*y^2 + 9*z^2 + x*y*z with x,y,z nonnegative integers.

This has been verified for all n = 0..10^6.

It seems that a(n) = 1 only for n = 0, 2, 5, 12, 65, 86, 155, 338, 21030.

Examples

			a(0) = 1 with 12*0 + 5 = 5 = 2*0^2 + 5*1^2 + 9*0^2 + 0*1*0.
a(2) = 1 with 12*2 + 5 = 29 = 2*0^2 + 5*2^2 + 9*1^2 + 0*2*1.
a(5) = 1 with 12*5 + 5 = 65 = 2*3^2 + 5*1^2 + 9*2^2 + 3*1*2.
a(12) = 1 with 12*12 + 5 = 149 = 2*0^2 + 5*1^2 + 9*4^2 + 0*1*4.
a(65) = 1 with 12*65 + 5 = 785 = 2*1^2 + 5*9^2 + 9*6^2 + 1*9*6.
a(86) = 1 with 12*86 + 5 = 1037 = 2*6^2 + 5*1^2 + 9*10^2 + 6*1*10.
a(155) = 1 with 12*155 + 5 = 1865 = 2*2^2 + 5*6^2 + 9*13^2 + 2*6*13.
a(338) = 1 with 12*338 + 5 = 4061 = 2*20^2 + 5*6^2 + 9*13^2 + 20*6*13.
a(21030) = 1 with 12*21030 + 5 = 252365 = 2*32^2 + 5*126^2 + 9*39^2 + 32*126*39.
		

Links

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

Crossrefs

Cf. A000290, A351723, A351617, A351902, A352259, A352286.

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[8(12n+5-5y^2-9z^2)+y^2*z^2]&&Mod[Sqrt[8(12n+5-5y^2-9z^2)+y^2*z^2]-y*z,4]==0,r=r+1],{y,0,Sqrt[(12n+5)/5]},{z,0,Sqrt[(12n+5-5y^2)/9]}];tab=Append[tab,r],{n,0,100}];Print[tab]