A033431 a(n) = 2*n^3.

0, 2, 16, 54, 128, 250, 432, 686, 1024, 1458, 2000, 2662, 3456, 4394, 5488, 6750, 8192, 9826, 11664, 13718, 16000, 18522, 21296, 24334, 27648, 31250, 35152, 39366, 43904, 48778, 54000, 59582, 65536, 71874, 78608, 85750, 93312, 101306, 109744, 118638, 128000, 137842

Offset: 0

Author

Jeff Burch

Keywords

• nonn
• easy

Comments

Also the largest determinant of a 3 X 3 matrix with entries from {0..n}. - Jud McCranie, Aug 12 2001

4*a(n) is a perfect cube.

The positive terms comprise the principal diagonal of the convolution array A213821. - Clark Kimberling, Jul 04 2012

Volume of a pyramid (square base) with side n and height 6*n. - Wesley Ivan Hurt, Aug 25 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Amelia Carolina Sparavigna, Generalized Sum of Stella Octangula Numbers, Politecnico di Torino (Italy, 2021).
• Amelia Carolina Sparavigna, Cardano Formula and Some Figurate Numbers, Politecnico di Torino (Italy, 2021).
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000578, A002117, A002378, A033581, A117642, A213821.

Programs

Magma

[2*n^3: n in [0..30]]; // Vincenzo Librandi, Jun 26 2011

Maple

seq(2*n^3, n=0..39); # Nathaniel Johnston, Jun 26 2011

Mathematica

2 Range[0, 50]^3 (* Wesley Ivan Hurt, Aug 25 2014 *)

PARI

a(n)=2*n^3 \\ Charles R Greathouse IV, Sep 24 2015

Formula

G.f.: 2*x*(1 + 4*x + x^2) / (1 - x)^4. - R. J. Mathar, Feb 04 2011

a(n) = 2*A000578(n). - Omar E. Pol, May 14 2008

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Aug 25 2014

a(n) = A002378(n)^2 - A002378(n^2). - Bruno Berselli, Oct 20 2016

E.g.f.: 2*x*(1 + 3*x + x^2)*exp(x). - G. C. Greubel, Jul 15 2017

From Amiram Eldar, Jan 10 2023: (Start)

Sum_{n>=1} 1/a(n) = zeta(3)/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = 3*zeta(3)/8. (End)