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

Author

N. J. A. Sloane, May 05 2004

Keywords

• nonn
• easy

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).

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.

Programs

GAP

List([0..50], n -> 3*n*(2*n-1)); # G. C. Greubel, Dec 07 2018

Magma

[3*n*(2*n-1): n in [0..50]]; // G. C. Greubel, Dec 07 2018

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 *)

PARI

a(n)=3*n*(2*n-1) \\ Charles R Greathouse IV, Sep 24 2015

Sage

[3*n*(2*n-1) for n in range(50)] # G. C. Greubel, Dec 07 2018

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

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Nov 16 2008

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