Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #41 Feb 28 2023 11:03:22
%S 0,5,30,91,204,385,650,1015,1496,2109,2870,3795,4900,6201,7714,9455,
%T 11440,13685,16206,19019,22140,25585,29370,33511,38024,42925,48230,
%U 53955,60116,66729,73810,81375,89440,98021,107134,116795,127020,137825,149226,161239,173880
%N a(n) = n*(2*n + 1)*(4*n + 1)/3.
%C First bisection of the square pyramidal numbers (A000330).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquarePyramidalNumber.html">Square Pyramidal Number</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: x*(5 + 10*x + x^2)/(1 - x)^4.
%F a(n) = A000330(2*n).
%F Sum_{n>0} 1/a(n) = 3*(6 - Pi - 4*log(2)) = 0.25745587...
%F 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
%F a(n) = A006918(4*n-1) = A053307(4*n-1) = A228706(4*n-1) for n>0. - _Bruno Berselli_, Nov 18 2015
%F 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
%F 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
%p A258582:=n->n*(2*n + 1)*(4*n + 1)/3: seq(A258582(n), n=0..50); # _Wesley Ivan Hurt_, Nov 17 2015
%t Table[(1/3) n (2 n + 1) (4 n + 1), {n, 0, 45}]
%o (PARI) vector(100, n, n--; n*(2*n+1)*(4*n+1)/3) \\ _Altug Alkan_, Nov 06 2015
%o (PARI) concat(0, Vec((5*x + 10*x^2 + x^3)/(1 - x)^4 + O(x^50))) \\ _Altug Alkan_, Nov 06 2015
%o (Magma) [n*(2*n+1)*(4*n+1)/3: n in [0..50]]; // _Wesley Ivan Hurt_, Nov 17 2015
%Y Cf. A000330, A001477, A005408, A016813, A053126 (partial sums), A100157.
%K nonn,easy
%O 0,2
%A _Ilya Gutkovskiy_, Nov 06 2015