%I #40 Mar 30 2023 02:37:14

%S 0,6,36,90,168,270,396,546,720,918,1140,1386,1656,1950,2268,2610,2976,

%T 3366,3780,4218,4680,5166,5676,6210,6768,7350,7956,8586,9240,9918,

%U 10620,11346,12096,12870,13668,14490,15336,16206,17100

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

%C 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

%C 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

%H Ivan Panchenko, <a href="/A152746/b152746.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

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

%F a(n) = a(n-1) + 24*n - 18 (with a(0)=0). - _Vincenzo Librandi_, Nov 26 2010

%F From _G. C. Greubel_, Sep 01 2018: (Start)

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

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

%F From _Amiram Eldar_, Mar 30 2023: (Start)

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

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

%t 6*PolygonalNumber[6,Range[0,40]] (* The program uses the PolygonalNumber function from Mathematica version 10 *) (* _Harvey P. Dale_, Mar 04 2016 *)

%t LinearRecurrence[{3,-3,1}, {0,6,36}, 50] (* or *) Table[6*n*(2*n-1), {n,0,50}] (* _G. C. Greubel_, Sep 01 2018 *)

%o (PARI) a(n)=6*n*(2*n-1) \\ _Charles R Greathouse IV_, Jun 17 2017

%o (Magma) [6*n*(2*n-1): n in [0..50]]; // _G. C. Greubel_, Sep 01 2018

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

%Y Column n=2 of A287318.

%K easy,nonn,walk

%O 0,2

%A _Omar E. Pol_, Dec 12 2008