A152746 Six times hexagonal numbers: 6*n*(2*n-1).

0, 6, 36, 90, 168, 270, 396, 546, 720, 918, 1140, 1386, 1656, 1950, 2268, 2610, 2976, 3366, 3780, 4218, 4680, 5166, 5676, 6210, 6768, 7350, 7956, 8586, 9240, 9918, 10620, 11346, 12096, 12870, 13668, 14490, 15336, 16206, 17100

Offset: 0

Author

Omar E. Pol, Dec 12 2008

Keywords

• easy
• nonn
• walk

Comments

Sequence found by reading the line from 0, in the direction 0, 6, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Sep 18 2011

a(n) is the number of walks on a cubic lattice of n dimensions that return to the origin, not necessarily for the first time, after 4 steps. - Shel Kaphan, Mar 20 2023

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, A001082, A002939, A094159, A085250, A152745.

Column n=2 of A287318.

Programs

Magma

[6*n*(2*n-1): n in [0..50]]; // G. C. Greubel, Sep 01 2018

Mathematica

6*PolygonalNumber[6,Range[0,40]] (* The program uses the PolygonalNumber function from Mathematica version 10 *) (* Harvey P. Dale, Mar 04 2016 *)
LinearRecurrence[{3,-3,1}, {0,6,36}, 50] (* or *) Table[6*n*(2*n-1), {n,0,50}] (* G. C. Greubel, Sep 01 2018 *)

PARI

a(n)=6*n*(2*n-1) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 12*n^2 - 6*n = A000384(n)*6 = A002939(n)*3 = A094159(n)*2.

a(n) = a(n-1) + 24*n - 18 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010

From G. C. Greubel, Sep 01 2018: (Start)

G.f.: 6*x*(1+3*x)/(1-x)^3.

E.g.f.: 6*x*(1+2*x)*exp(x). (End)

From Amiram Eldar, Mar 30 2023: (Start)

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

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/12 - log(2)/6. (End)