A152750 Eight times hexagonal numbers: a(n) = 8*n*(2*n-1).

0, 8, 48, 120, 224, 360, 528, 728, 960, 1224, 1520, 1848, 2208, 2600, 3024, 3480, 3968, 4488, 5040, 5624, 6240, 6888, 7568, 8280, 9024, 9800, 10608, 11448, 12320, 13224, 14160, 15128, 16128, 17160, 18224, 19320, 20448, 21608, 22800, 24024, 25280, 26568, 27888, 29240

Offset: 0

Author

Omar E. Pol, Dec 12 2008

Keywords

• easy
• nonn

Comments

Equals Engel expansion of cosh(1/2), except first member (see A067239).

Also sequence found by reading the line from 0, in the direction 0, 8, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Sep 18 2011

a(n) = the sum of the edges of a rectangular prism having edges 2*(n-1)*n, n^2 - (n-1)^2, and n^2 + (n-1)^2. - J. M. Bergot, Apr 24 2014

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000384, A002939, A067239, A074377, A085250.

Programs

Magma

[ 8*n*(2*n-1) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

Maple

A152750:=n->8*n*(2*n-1); seq(A152750(n), n=0..50); # Wesley Ivan Hurt, Jun 09 2014

Mathematica

Table[8*n*(2*n - 1), {n, 0, 50}] (* Wesley Ivan Hurt, Jun 09 2014 *)

PARI

concat(0, Vec(8*x*(1+3*x)/(1-x)^3 + O(x^50))) \\ Colin Barker, Sep 25 2016

Formula

a(n) = 16*n^2 - 8*n = 8*A000384(n) = 4*A002939(n) = 2*A085250(n).

a(n) = A067239(n), for n > 0.

a(n) = a(n-1) + 32*n - 24 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010

From Colin Barker, Sep 25 2016: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.

G.f.: 8*x*(1+3*x)/(1-x)^3. (End)

Sum_{n>=1} 1/a(n) = log(2)/4. - Vaclav Kotesovec, Sep 25 2016

E.g.f.: 8*exp(x)*x*(1 + 2*x). - Elmo R. Oliveira, Dec 15 2024

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/16 - log(2)/8. - Amiram Eldar, May 05 2025