A356251 - OEIS

%I #28 Sep 11 2022 06:32:35

%S 0,6,50,210,630,1540,3276,6300,11220,18810,30030,46046,68250,98280,

%T 138040,189720,255816,339150,442890,570570,726110,913836,1138500,

%U 1405300,1719900,2088450,2517606,3014550,3587010,4243280,4992240,5843376,6806800,7893270,9114210

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

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

%H Edward Krogius, <a href="/A356251/b356251.txt">Table of n, a(n) for n = 0..999</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

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

%F From _Amiram Eldar_, Sep 11 2022: (Start)

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

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

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

%t Table[n*(n + 1)*(n + 2)*(n + 3)*(2*n + 1)/12, {n, 0, 35}] (* _Amiram Eldar_, Sep 11 2022 *)

%Y Cf. A033487.

%K nonn,easy

%O 0,2

%A _Edward Krogius_, Jul 31 2022