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

Author

Peter M. Chema, Jan 09 2016

Keywords

• nonn
• easy
• changed

Comments

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

Examples

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

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).

Crossrefs

Cf. A000292, A000447, A005843, A069140.

Programs

Magma

[32/3*n^3-2/3*n: n in [0..35]]; // Vincenzo Librandi, Jan 10 2016

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 *)

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

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

From Elmo R. Oliveira, Sep 06 2025: (Start)

E.g.f.: 2*x*(15 + 48*x + 16*x^2)*exp(x)/3.

a(n) = A069140(n)/6. (End)

Extensions

More terms from Michael De Vlieger, Jan 09 2016

First term added from Vincenzo Librandi, Jan 10 2016