A351902 Number of ways to write n as w^2 + x^2 + y^2 + z^2 + 3*x*y*z, where w is a positive integer, and x,y,z are nonnegative integers with x <= y <= z.

1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 3, 3, 1, 2, 4, 3, 2, 2, 5, 4, 0, 3, 4, 5, 3, 1, 6, 4, 2, 1, 5, 5, 3, 5, 5, 5, 1, 3, 8, 4, 3, 2, 7, 7, 1, 3, 5, 7, 5, 3, 5, 9, 3, 4, 8, 3, 5, 1, 9, 8, 1, 2, 8, 9, 3, 5, 9, 6, 2, 5, 6, 8, 4, 6, 7, 7, 1, 3, 15, 6, 5, 5, 9, 9, 2, 4, 12, 9, 5, 2, 5, 10, 1, 5, 9, 8, 7, 5

Offset: 1

Author

Zhi-Wei Sun, Feb 25 2022

Keywords

• nonn

Comments

Conjecture: We have a(n) > 0 except for n = 23. In other words, any positive integer n not equal to 23 can be written as w^2 + x^2 + y^2 + z^2 + 3*x*y*z, where w is a positive integer and x,y,z are nonnegative integers.

It seems that a(n) > 1 for all n > 695.

Examples

			a(60)  = 1 with  60 =  2^2 + 1^2 + 1^2 +  6^2 + 3*1*1*6.
a(128) = 1 with 128 =  8^2 + 0^2 + 0^2 +  8^2 + 3*0*0*8.
a(303) = 1 with 303 = 11^2 + 1^2 + 1^2 + 12^2 + 3*1*1*12.
a(359) = 1 with 359 =  3^2 + 1^2 + 5^2 + 12^2 + 3*1*5*12.
a(383) = 1 with 383 = 11^2 + 1^2 + 3^2 + 12^2 + 3*1*3*12.
a(695) = 1 with 695 = 17^2 + 1^2 + 9^2 +  9^2 + 3*1*9*9.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000118, A000290, A000378, A351723.

Programs

Mathematica

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