A015222 Even square pyramidal numbers.

14, 30, 140, 204, 506, 650, 1240, 1496, 2470, 2870, 4324, 4900, 6930, 7714, 10416, 11440, 14910, 16206, 20540, 22140, 27434, 29370, 35720, 38024, 45526, 48230, 56980, 60116, 70210, 73810, 85344, 89440, 102510, 107134, 121836, 127020, 143450, 149226, 167480, 173880

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..40.

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Formula

Even entries in A000330.

From Artur Jasinski, Oct 22 2008: (Start)

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

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

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]

From Ant King, Oct 17 2012: (Start)

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

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

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

From Amiram Eldar, Mar 07 2022: (Start)

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

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

Mathematica

Select[ Table[ n(n+1)(2n+1)/6, {n, 100} ], EvenQ ]
Select[Rest[CoefficientList[Series[(x(x+1))/(x-1)^4, {x, 0, 80}], x]], EvenQ]  (* Harvey P. Dale, May 05 2011 *)

Crossrefs

Cf. A000330, A015221.

Sequence in context: A075208 A228124 A293391 * A054103 A344397 A161454

Adjacent sequences: A015219 A015220 A015221 * A015223 A015224 A015225

Keyword

nonn,easy

Author

Mohammad K. Azarian

Extensions

More terms from Erich Friedman

Status

approved