A304170 a(n) = 32*3^n + 18*2^n - 116 (n>=1).

16, 244, 892, 2764, 8236, 24364, 72172, 214444, 638956, 1907884, 5705452, 17079724, 51165676, 153349804, 459754732, 1378674604, 4134844396, 12402174124, 37201804012, 111595975084, 334769051116, 1004269404844, 3012732717292, 9038047157164, 27113839481836, 81340914465964, 244021535438572, 732062190396844

Offset: 0

Author

Emeric Deutsch, May 11 2018

Keywords

• nonn
• easy

Comments

For n>=2, a(n) is the second Zagreb index of the Sierpinski Gasket Rhombus graph SR[n] (see the Antony Xavier et al. reference).

The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph.

The M-polynomial of the Sierpinski Gasket Rhombus graph SR[n] is M(SR[n]; x,y) = 4*x^2*y^4 + 4*x^3*y^4 +2*x^3*y^6 + (2*3^n - 3*2^n - 4)*x^4*y^4 + (2^{n+1} - 4)*x^4*y^6 + (2^{n-1} - 2)*x^6*y^6.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
• D. Antony Xavier, M. Rosary, and Andrew Arokiaraj, Topological properties of Sierpinski Gasket Rhombus graphs, International J. of Mathematics and Soft Computing, 4, No. 2, 2014, 95-104.
• Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Crossrefs

Cf. A304169.

Programs

Maple

seq(32*3^n+18*2^n-116, n = 1 .. 40);

Mathematica

CoefficientList[Series[4*(4 + 37*x - 99*x^2)/((1 - x)*(1 - 2*x)*(1 - 3*x)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jan 20 2024 *)
LinearRecurrence[{6,-11,6},{16,244,892},30] (* Harvey P. Dale, Feb 13 2024 *)

PARI

Vec(4*(4 + 37*x - 99*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, May 12 2018

Formula

From Colin Barker, May 12 2018: (Start)

G.f.: 4*(4 + 37*x - 99*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)).

a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>2.

(End)