A304169 a(n) = 16*3^n + 2^(n+1) - 26 (n>=1).

26, 126, 422, 1302, 3926, 11766, 35222, 105462, 315926, 946806, 2838422, 8511222, 25525526, 76560246, 229648022, 688878582, 2066504726, 6199252086, 18597232022, 55790647542, 167369845526, 502105342326, 1506307638422, 4518906138102, 13556684859926, 40669987470966

Offset: 1

Author

Emeric Deutsch, May 11 2018

Keywords

• nonn
• easy

Comments

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

The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, 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 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 = 1..1000
• 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.
• E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
• Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Crossrefs

Cf. A304170.

Programs

Maple

seq(16*3^n+2^(n+1)-26, n = 1 .. 30);

PARI

Vec(2*x*(13 - 15*x - 24*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, May 11 2018

Formula

From Colin Barker, May 11 2018: (Start)

G.f.: 2*x*(13 - 15*x - 24*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>3.

(End)