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

0,2

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).

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)

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 *)

Prog

(PARI) a(n)=6*n*(2*n-1) \\ Charles R Greathouse IV, Jun 17 2017
(Magma) [6*n*(2*n-1): n in [0..50]]; // G. C. Greubel, Sep 01 2018

Crossrefs

Cf. A000384, A001082, A002939, A094159, A085250, A152745.

Column n=2 of A287318.

Sequence in context: A207896 A207656 A207341 * A207363 A207600 A207026

Adjacent sequences: A152743 A152744 A152745 * A152747 A152748 A152749

Keyword

easy,nonn,walk

Author

Omar E. Pol, Dec 12 2008

Status

approved