A152743 6 times pentagonal numbers: a(n) = 3*n*(3*n-1).

0, 6, 30, 72, 132, 210, 306, 420, 552, 702, 870, 1056, 1260, 1482, 1722, 1980, 2256, 2550, 2862, 3192, 3540, 3906, 4290, 4692, 5112, 5550, 6006, 6480, 6972, 7482, 8010, 8556, 9120, 9702, 10302, 10920, 11556, 12210, 12882, 13572, 14280, 15006, 15750, 16512, 17292

Offset: 0

Author

Omar E. Pol, Dec 12 2008

Keywords

• easy
• nonn

Comments

a(n) is also the Wiener index of the windmill graph D(4,n). The windmill graph D(m,n) is the graph obtained by taking n copies of the complete graph K_m with a vertex in common (i.e. a bouquet of n pieces of K_m graphs). The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices in the graph. The Wiener index of D(m,n) is (1/2)n(m-1)[(m-1)(2n-1)+1]. For the Wiener indices of D(3,n), D(5,n), and D(6,n) see A033991, A028994, and A180577, respectively. - Emeric Deutsch, Sep 21 2010

a(n+1) gives the number of edges in a hexagon-like honeycomb built from A003215(n) congruent regular hexagons (see link). Example: a hexagon-like honeycomb consisting of 7 congruent regular hexagons has 1 core hexagon inside a perimeter of six hexagons. The perimeter consists of 18 external edges. There are 6 edges shared by the perimeter hexagons. The core hexagon has 6 edges. a(2) is the total number of edges, i.e. 18 + 6 + 6 = 30. - Ivan N. Ianakiev, Mar 10 2015

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Ivan N. Ianakiev, Hexagon-like honeycomb built from regular congruent hexagons.
• Eric Weisstein's World of Mathematics, Windmill Graph.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000326, A152734, A152744, A049450, A062741, A033991, A028994, A180577.

Programs

Magma

[ 3*n*(3*n-1) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

Maple

A152743:=n->3*n*(3*n-1); seq(A152743(n), n=0..50); # Wesley Ivan Hurt, Jun 09 2014

Mathematica

Table[3n(3n-1),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,6,30},40] (* Harvey P. Dale, Jun 30 2011 *)
CoefficientList[Series[-6x (2x+1)/(x-1)^3,{x,0,40}],x] (* Robert G. Wilson v, Mar 10 2015 *)

PARI

a(n)=3*n*(3*n-1) \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = 9n^2 - 3n = A000326(n)*6.

a(n) = A049450(n)*3 = A062741(n)*2. - Omar E. Pol, Dec 15 2008

a(n) = a(n-1) + 18*n - 12 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010

G.f.: -((6*x*(2*x+1))/(x-1)^3). - Harvey P. Dale, Jun 30 2011

E.g.f.: 3*x*(2+3*x)*exp(x). - G. C. Greubel, Sep 01 2018

From Amiram Eldar, Feb 27 2022: (Start)

Sum_{n>=1} 1/a(n) = (9*log(3) - sqrt(3)*Pi)/18.

Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi*sqrt(3) - 6*log(2))/9. (End)

Extensions

Converted reference to link by Omar E. Pol, Oct 07 2010