A267031 a(n) = (32*n^3 - 2*n)/3.

0, 10, 84, 286, 680, 1330, 2300, 3654, 5456, 7770, 10660, 14190, 18424, 23426, 29260, 35990, 43680, 52394, 62196, 73150, 85320, 98770, 113564, 129766, 147440, 166650, 187460, 209934, 234136, 260130, 287980, 317750, 349504, 383306, 419220, 457310, 497640, 540274, 585276, 632710, 682640, 735130, 790244

Offset

0,2

Comments

This sequence alternates with the tetrahedral numbers, A000292, to create the centered octagonal pyramidal number sequence, A000447.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

G.f.: 2*x*(5 + 22*x + 5*x^2)/(-1 + x)^4. - Michael De Vlieger, Jan 09 2016

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - Colin Barker, Jan 10 2016

From Amiram Eldar, Jan 04 2022: (Start)

Sum_{n>=1} 1/a(n) = 9*log(2)/2 - 3.

Sum_{n>=1} (-1)^(n+1)/a(n) = 3 - 3*(2-sqrt(2))*log(2)/4 - 3*sqrt(2)*log(sqrt(2)+2)/2. (End)

a(n) = binomial(4*n+1, 3). - Michel Marcus, Mar 05 2022

a(n) = 8*A000447(n) + A005843(n). - Yasser Arath Chavez Reyes, Mar 02 2024

Example

a(4) = (32/3)*4^3 - (2/3)*4 = 680.

Mathematica

Table[(32 n^3 - 2 n)/3, {n, 0, 42}] (* or *)
CoefficientList[Series[(2 x (5 + 22 x + 5 x^2))/(-1 + x)^4, {x, 0, 41}], x] (* Michael De Vlieger, Jan 09 2016 *)

Prog

(Magma) [32/3*n^3-2/3*n: n in [0..35]]; // Vincenzo Librandi, Jan 10 2016
(PARI) concat(0, Vec(2*x*(5+22*x+5*x^2)/(1-x)^4 + O(x^100))) \\ Colin Barker, Jan 10 2016
(PARI) a(n) = (32*n^3 - 2*n)/3; \\ Altug Alkan, Jan 10 2015

Crossrefs

Cf. A000292, A000447.

Sequence in context: A226202 A271557 A351750 * A289163 A092718 A090763

Adjacent sequences: A267028 A267029 A267030 * A267032 A267033 A267034

Keyword

nonn,easy

Author

Peter M. Chema, Jan 09 2016

Extensions

More terms from Michael De Vlieger, Jan 09 2016

First term added from Vincenzo Librandi, Jan 10 2016

Status

approved