A277106 a(n) = 8*3^n - 12.

12, 60, 204, 636, 1932, 5820, 17484, 52476, 157452, 472380, 1417164, 4251516, 12754572, 38263740, 114791244, 344373756, 1033121292, 3099363900, 9298091724, 27894275196, 83682825612, 251048476860, 753145430604, 2259436291836, 6778308875532

Offset: 1

Author

Emeric Deutsch, Nov 05 2016

Keywords

• nonn
• easy

Comments

a(n) is the first Zagreb index of the Sierpiński [Sierpinski] gasket graph S[n].

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.

The M-polynomial of the Sierpinski gasket graph S[n] is M(S[n],x,y) = 6*x^2*y^4 + (3^n - 6)*x^4*y^4.

Links

• E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
• I. Gutman and K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.
• Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph.
• Index entries for linear recurrences with constant coefficients, signature (4,-3).

Crossrefs

Cf. A277107.

Programs

Maple

seq(8*3^n-12, n = 1..30);

Mathematica

Array[8*3^# - 12 &, 25] (* Robert G. Wilson v, Nov 05 2016 *)
LinearRecurrence[{4,-3},{12,60},40] (* Harvey P. Dale, Oct 25 2020 *)

Formula

G.f.: 12*x*(1 + x)/((1 - x)*(1 - 3*x)).

a(n) = 4*a(n-1) - 3*a(n-2).

a(n)=12*A048473(n-1). - R. J. Mathar, Apr 07 2022