A094159 3 times hexagonal numbers: a(n) = 3*n*(2*n-1).

0, 3, 18, 45, 84, 135, 198, 273, 360, 459, 570, 693, 828, 975, 1134, 1305, 1488, 1683, 1890, 2109, 2340, 2583, 2838, 3105, 3384, 3675, 3978, 4293, 4620, 4959, 5310, 5673, 6048, 6435, 6834, 7245, 7668, 8103, 8550, 9009, 9480, 9963, 10458, 10965, 11484

Offset

0,2

Comments

Column 3 of A048790.

Sequence found by reading the line from 0, in the direction 0, 3, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011

a(n) is the sum of all perimeters of triangles having two sides of length n. For n=4 one has seven triangles with two sides of length 4 and the other of lengths 1..7. - J. M. Bergot, Mar 26 2014

a(n) is the Wiener index of the complete tripartite graph K_{n,n,n}. - Eric W. Weisstein, Sep 07 2017

Sequence found by reading the line from 0, in the direction 0, 3, ..., in a spiral on an equilateral triangular lattice. - Hans G. Oberlack, Dec 08 2018

References

Dan Hoey, Bill Gosper and Richard C. Schroeppel, Discussions in Math-Fun Mailing list, circa Jul 13 1999.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Hans G. Oberlack, Triangle spiral.

R. C. Schroeppel, A few mathematical experiments, Experimental Mathematics Workshop, Oakland, California, March 30, 2004.

Eric Weisstein's World of Mathematics, Complete Tripartite Graph.

Eric Weisstein's World of Mathematics, Wiener Index.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = 6*n^2 - 3*n = 3*n*(2*n-1) = 3*A000384(n). - Omar E. Pol, Dec 11 2008

a(n) = 12*n + a(n-1) - 9 with n > 0, a(0)=0. - Vincenzo Librandi, Nov 16 2010

G.f.: 3*x*(1+3*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011

Sum_{n>0} 1/a(n) = (2/3)*log(2). - Enrique Pérez Herrero, Jun 04 2015

E.g.f.: 3*x*(1+2*x)*exp(x). - G. C. Greubel, Dec 07 2018

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/6 - log(2)/3. - Amiram Eldar, Jan 10 2022

Maple

A094159:=n->3*n*(2*n-1); seq(A094159(n), n=0..40); # Wesley Ivan Hurt, Mar 28 2014

Mathematica

CoefficientList[Series[3x(1+3x)/(1-x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 19 2013 *)
Table[3n(2n-1), {n, 0, 50}] (* or *) 3*PolygonalNumber[6, Range[0, 50]] (* or *) LinearRecurrence[{3, -3, 1}, {3, 18, 45}, {0, 50}] (* Eric W. Weisstein, Sep 07 2017 *)

Prog

(PARI) a(n)=3*n*(2*n-1) \\ Charles R Greathouse IV, Sep 24 2015
(Magma) [3*n*(2*n-1): n in [0..50]]; // G. C. Greubel, Dec 07 2018
(Sage) [3*n*(2*n-1) for n in range(50)] # G. C. Greubel, Dec 07 2018
(GAP) List([0..50], n -> 3*n*(2*n-1)); # G. C. Greubel, Dec 07 2018

Crossrefs

Cf. A000384, A048790.

3 times n-gonal numbers: A045943, A033428, A062741, A152773, A152751, A152759, A152767, A153783, A153448, A153875.

Essentially a bisection of A045943. - Omar E. Pol, Sep 17 2011

Cf. numbers of the form n*(n*k-k+6))/2, this sequence is the case k=12: see Comments lines of A226492.

Sequence in context: A069147 A337921 A365442 * A138976 A275038 A304976

Adjacent sequences: A094156 A094157 A094158 * A094160 A094161 A094162

Keyword

nonn,easy

Author

N. J. A. Sloane, May 05 2004

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Nov 16 2008

Definition improved, offset corrected and edited by Omar E. Pol, Dec 11 2008

Status

approved