OFFSET
0,2
COMMENTS
First bisection of the square pyramidal numbers (A000330).
LINKS
Eric Weisstein's World of Mathematics, Square Pyramidal Number.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
G.f.: x*(5 + 10*x + x^2)/(1 - x)^4.
a(n) = A000330(2*n).
Sum_{n>0} 1/a(n) = 3*(6 - Pi - 4*log(2)) = 0.25745587...
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Vincenzo Librandi, Nov 18 2015
a(n) = Sum_{k=1..2*n} k^2 (see the first comment). E.g.f. exp(x)*(5*x+ 20*x^2/2+16*x^3/3!). - Wolfdieter Lang, Mar 13 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2) + 6*sqrt(2)*log(1+sqrt(2)) + 3*(sqrt(2)-1/2)*Pi - 18. - Amiram Eldar, Sep 17 2022
MAPLE
MATHEMATICA
Table[(1/3) n (2 n + 1) (4 n + 1), {n, 0, 45}]
PROG
(PARI) vector(100, n, n--; n*(2*n+1)*(4*n+1)/3) \\ Altug Alkan, Nov 06 2015
(PARI) concat(0, Vec((5*x + 10*x^2 + x^3)/(1 - x)^4 + O(x^50))) \\ Altug Alkan, Nov 06 2015
(Magma) [n*(2*n+1)*(4*n+1)/3: n in [0..50]]; // Wesley Ivan Hurt, Nov 17 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Nov 06 2015
STATUS
approved