%I #43 Aug 04 2024 18:57:52

%S 14,30,140,204,506,650,1240,1496,2470,2870,4324,4900,6930,7714,10416,

%T 11440,14910,16206,20540,22140,27434,29370,35720,38024,45526,48230,

%U 56980,60116,70210,73810,85344,89440,102510,107134,121836,127020,143450,149226,167480,173880

%N Even square pyramidal numbers.

%C Square pyramidal numbers k*(k + 1)*(2*k + 1)/6 are even if and only when k is congruent to 0 or 3 mod 4. - _Artur Jasinski_, Oct 22 2008

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).

%F Even entries in A000330.

%F From _Artur Jasinski_, Oct 22 2008: (Start)

%F (2*k + 1)/(k + 2)*binomial(k + 2, 5) if k congruent to 0 or 3 mod 4, and

%F k*(k + 1)*(2*k + 1)/6 if k congruent to 0 or 3 mod 4. (End)

%F G.f.: 2*x*(7+x*(8+x*(34+x*(8+7*x)))) / ((-1+x)^4*(1+x)^3). - _Harvey P. Dale_, May 05 2011 [adapted to the offset by _Bruno Berselli_, May 16 2011]

%F From _Ant King_, Oct 17 2012: (Start)

%F a(n) = (3 + 4*n - (-1)^n)*(2 + 4*n - (-1)^n)*(1 + 4*n - (-1)^n)/24.

%F a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) + 128. (End)

%F 6*a(n) = (1+2*n)*(8*n^2+8*n-6*(-1)^n*n+3-3*(-1)^n). - _R. J. Mathar_, Oct 17 2012

%F From _Amiram Eldar_, Mar 07 2022: (Start)

%F Sum_{n>=1} 1/a(n) = 18 + 6*sqrt(2)*log(2-sqrt(2)) - 3*(sqrt(2)+5)*log(2).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi*(sqrt(2)+1/2) - 18. (End)

%t Select[ Table[ n(n+1)(2n+1)/6, {n, 100} ], EvenQ ]

%t Select[Rest[CoefficientList[Series[(x(x+1))/(x-1)^4,{x,0,80}], x]],EvenQ] (* _Harvey P. Dale_, May 05 2011 *)

%Y Cf. A000330, A015221.

%K nonn,easy,changed

%O 1,1

%A _Mohammad K. Azarian_

%E More terms from _Erich Friedman_