

A153792


12 times pentagonal numbers: a(n) = 6*n*(3*n1).


3



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
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OFFSET

0,2


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 Mpolynomial of the triangular grid graph T[n] is M(T[n], x, y) = 6*x^2*y^4 + 3*(n1)*x^4*y^4 +6*(n2)*x^4*y^6+3*(n2)*(n3)*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 11gonal numbers A195160.  Omar E. Pol, Mar 12 2021


REFERENCES

D. B. West, Introduction to Graph Theory, 2nd edition, PrenticeHall, 2001.


LINKS

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Emeric Deutsch and Sandi Klavzar, Mpolynomial and degreebased topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93102.
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).


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(n1)  24 (with a(0)=0).  Vincenzo Librandi, Aug 03 2010
G.f.: 12*x*(1 + 2*x)/(1x)^3.  Colin Barker, Feb 14 2012
a(0)=0, a(1)=12, a(2)=60; for n>2, a(n) = 3*a(n1)  3*a(n2) + a(n3).  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


MAPLE

seq(6*n*(3*n1), n=0..50); # Robert Israel, Nov 10 2016


MATHEMATICA

Table[6n(3n1), {n, 0, 40}] (* or *) LinearRecurrence[{3, 3, 1}, {0, 12, 60}, 40] (* Harvey P. Dale, Mar 11 2012 *)


PROG

(PARI) a(n)=6*n*(3*n1) \\ Charles R Greathouse IV, Jun 17 2017
(GAP) List([0..50], n>6*n*(3*n1)); # Muniru A Asiru, May 10 2018


CROSSREFS

Cf. A000326, A049450, A062741, A033579, A152743, A153449, A153793.
Cf. A008588, A195160, A195321.
Sequence in context: A099829 A099830 A158443 * A229616 A321465 A000141
Adjacent sequences: A153789 A153790 A153791 * A153793 A153794 A153795


KEYWORD

nonn,easy


AUTHOR

Omar E. Pol, Jan 01 2009


STATUS

approved



