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%I #71 Mar 18 2021 23:58:09
%S 0,12,60,144,264,420,612,840,1104,1404,1740,2112,2520,2964,3444,3960,
%T 4512,5100,5724,6384,7080,7812,8580,9384,10224,11100,12012,12960,
%U 13944,14964,16020,17112,18240,19404,20604,21840,23112,24420
%N 12 times pentagonal numbers: a(n) = 6*n*(3*n-1).
%C 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
%C 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
%C This is the number of overlapping six sphinx tiled shapes in the sphinx tessellated hexagon described in A291582. - _Craig Knecht_, Sep 13 2017
%C 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
%C 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
%D D. B. West, Introduction to Graph Theory, 2nd edition, Prentice-Hall, 2001.
%H Harvey P. Dale, <a href="/A153792/b153792.txt">Table of n, a(n) for n = 0..1000</a>
%H Emeric Deutsch and Sandi Klavzar, <a href="http://dx.doi.org/10.22052/ijmc.2015.10106">M-polynomial and degree-based topological indices</a>, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
%H Craig Knecht, <a href="/A153792/a153792.png">Example of 12 overlapping shapes in the order 1 hexagon.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularGridGraph.html">Triangular Grid Graph</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 18*n^2 - 6*n = 12*A000326(n) = 6*A049450(n) = 4*A062741(n) = 3*A033579(n) = 2*A152743(n).
%F a(n) = 36*n + a(n-1) - 24 (with a(0)=0). - _Vincenzo Librandi_, Aug 03 2010
%F G.f.: 12*x*(1 + 2*x)/(1-x)^3. - _Colin Barker_, Feb 14 2012
%F 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
%F E.g.f.: 6*x*(2 + 3*x)*exp(x). - _G. C. Greubel_, Aug 29 2016
%F a(n) = A291582(n) - A195321(n) for n > 0. - _Craig Knecht_, Sep 13 2017
%F a(n) = A195321(n) - A008588(n). - _Omar E. Pol_, Mar 12 2021
%p seq(6*n*(3*n-1),n=0..50); # _Robert Israel_, Nov 10 2016
%t Table[6n(3n-1),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,12,60},40] (* _Harvey P. Dale_, Mar 11 2012 *)
%o (PARI) a(n)=6*n*(3*n-1) \\ _Charles R Greathouse IV_, Jun 17 2017
%o (GAP) List([0..50],n->6*n*(3*n-1)); # _Muniru A Asiru_, May 10 2018
%Y Cf. A000326, A049450, A062741, A033579, A152743, A153449, A153793.
%Y Cf. A008588, A195160, A195321.
%K nonn,easy
%O 0,2
%A _Omar E. Pol_, Jan 01 2009
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