A153792 12 times pentagonal numbers: a(n) = 6*n*(3*n-1).

0, 12, 60, 144, 264, 420, 612, 840, 1104, 1404, 1740, 2112, 2520, 2964, 3444, 3960, 4512, 5100, 5724, 6384, 7080, 7812, 8580, 9384, 10224, 11100, 12012, 12960, 13944, 14964, 16020, 17112, 18240, 19404, 20604, 21840, 23112, 24420

Offset: 0

Author

Omar E. Pol, Jan 01 2009

Keywords

• nonn
• easy

Comments

For n>=1, a(n) is the first Zagreb index of the triangular grid graph T[n] (see the West reference, p. 390). The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph. - Emeric Deutsch, Nov 10 2016

The M-polynomial of the triangular grid graph T[n] is M(T[n], x, y) = 6*x^2*y^4 + 3*(n-1)*x^4*y^4 +6*(n-2)*x^4*y^6+3*(n-2)*(n-3)*x^6*y^6/2. - Emeric Deutsch, May 09 2018

This is the number of overlapping six sphinx tiled shapes in the sphinx tessellated hexagon described in A291582. - Craig Knecht, Sep 13 2017

a(n) is the number of words of length 3n over the alphabet {a,b,c}, where the number of b's plus the number of c's is 2. - Juan Camacho, Mar 03 2021

Sequence found by reading the line from 0, in the direction 0, 12, ..., in the square spiral whose vertices are the generalized 11-gonal numbers A195160. - Omar E. Pol, Mar 12 2021

References

• D. B. West, Introduction to Graph Theory, 2nd edition, Prentice-Hall, 2001.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
• Craig Knecht, Example of 12 overlapping shapes in the order 1 hexagon.
• Eric Weisstein's World of Mathematics, Triangular Grid Graph.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000326, A049450, A062741, A033579, A152743, A153449, A153793.

Cf. A008588, A195160, A195321.

Programs

GAP

List([0..50],n->6*n*(3*n-1)); # Muniru A Asiru, May 10 2018

Maple

seq(6*n*(3*n-1),n=0..50); # Robert Israel, Nov 10 2016

Mathematica

Table[6n(3n-1),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,12,60},40] (* Harvey P. Dale, Mar 11 2012 *)

PARI

a(n)=6*n*(3*n-1) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 18*n^2 - 6*n = 12*A000326(n) = 6*A049450(n) = 4*A062741(n) = 3*A033579(n) = 2*A152743(n).

a(n) = 36*n + a(n-1) - 24 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010

G.f.: 12*x*(1 + 2*x)/(1-x)^3. - Colin Barker, Feb 14 2012

a(0)=0, a(1)=12, a(2)=60; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Mar 11 2012

E.g.f.: 6*x*(2 + 3*x)*exp(x). - G. C. Greubel, Aug 29 2016

a(n) = A291582(n) - A195321(n) for n > 0. - Craig Knecht, Sep 13 2017

a(n) = A195321(n) - A008588(n). - Omar E. Pol, Mar 12 2021

From Amiram Eldar, May 05 2025: (Start)

Sum_{n>=1} 1/a(n) = log(3)/4 - Pi/(12*sqrt(3)).

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