A356251 a(n) = n*(n+1)*(n+2)*(n+3)*(2*n+1)/12.

0, 6, 50, 210, 630, 1540, 3276, 6300, 11220, 18810, 30030, 46046, 68250, 98280, 138040, 189720, 255816, 339150, 442890, 570570, 726110, 913836, 1138500, 1405300, 1719900, 2088450, 2517606, 3014550, 3587010, 4243280, 4992240, 5843376, 6806800, 7893270, 9114210

Offset: 0

Author

Edward Krogius, Jul 31 2022

Keywords

• nonn
• easy

Comments

Sum of all numbers squared in ordered triples (x,y,z) where 0 <= x <= y <= z <= n.

Examples

			a(1) = 6 because we have the triples (0,0,0), (0,0,1), (0,1,1), (1,1,1).
		

Links

• Edward Krogius, Table of n, a(n) for n = 0..999
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A033487.

Programs

Mathematica

Table[n*(n + 1)*(n + 2)*(n + 3)*(2*n + 1)/12, {n, 0, 35}] (* Amiram Eldar, Sep 11 2022 *)
Table[Sum[x^2 + y^2 + z^2, {x, 0, g}, {y, x, g}, {z, y, g}], {g, 0, 30}] (* Horst H. Manninger, Jun 19 2025 *)

Formula

G.f.: 2*x*(7*x+3)/(x-1)^6.

From Amiram Eldar, Sep 11 2022: (Start)

Sum_{n>=1} 1/a(n) = 136/15 - 64*log(2)/5.

Sum_{n>=1} (-1)^(n+1)/a(n) = 16*Pi/5 - 32*log(2)/5 - 82/15. (End)