A354777 Irregular triangle read by rows: T(n,k) is the number of integer quadruples (u,v,w,x) such that u^2+v^2+w^2+x^2 = n and u+v+w+x = k (n>=0, 0 <= k <= A307531(n)).

1, 0, 4, 12, 0, 6, 0, 12, 0, 4, 6, 0, 8, 0, 1, 0, 12, 0, 12, 24, 0, 24, 0, 12, 0, 16, 0, 12, 0, 4, 12, 0, 0, 0, 6, 0, 24, 0, 16, 0, 12, 24, 0, 30, 0, 24, 0, 6, 0, 12, 0, 24, 0, 12, 8, 0, 24, 0, 12, 0, 8, 0, 24, 0, 12, 0, 16, 0, 4, 48, 0, 24, 0, 24, 0, 24, 0, 36, 0, 24, 0, 24, 0, 12, 6, 0, 0, 0, 8, 0, 0, 0, 1, 0, 12, 0, 36, 0, 12, 0, 12

Offset: 0

Author

N. J. A. Sloane, Jun 27 2022

Keywords

• nonn
• tabf

Comments

Row n has width A307531(n).

Examples

			The triangle begins:
[1],
[0, 4],
[12, 0, 6],
[0, 12, 0, 4],
[6, 0, 8, 0, 1],
[0, 12, 0, 12],
[24, 0, 24, 0, 12],
[0, 16, 0, 12, 0, 4],
[12, 0, 0, 0, 6],
[0, 24, 0, 16, 0, 12],
[24, 0, 30, 0, 24, 0, 6],
[0, 12, 0, 24, 0, 12],
[8, 0, 24, 0, 12, 0, 8],
[0, 24, 0, 12, 0, 16, 0, 4],
[48, 0, 24, 0, 24, 0, 24],
[0, 36, 0, 24, 0, 24, 0, 12],
[6, 0, 0, 0, 8, 0, 0, 0, 1],
[0, 12, 0, 36, 0, 12, 0, 12],
[36, 0, 48, 0, 48, 0, 30, 0, 12],
...
T(4,2) = 8 from the solutions (u,v,w,x) = (2,0,0,0) (4 such) and (1,1,1,-1) (4 such).
		

Crossrefs

Cf. A000118, A307531.

T(n^2,n) = A354778(n). See also A278085 and A354766.