A304167 - OEIS

A304167

a(n) = 3^n - 2^(n-1) + 2 (n>=1).

4, 9, 25, 75, 229, 699, 2125, 6435, 19429, 58539, 176125, 529395, 1590229, 4774779, 14332525, 43013955, 129074629, 387289419, 1161999325, 3486260115, 10459304629, 31378962459, 94138984525, 282421147875, 847271832229, 2541832273899, 7625530376125, 22876658237235, 68630108929429, 205890595223739

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OFFSET

1,1

COMMENTS

For n>=2, a(n) is the number of vertices of the Sierpinski Gasket Rhombus graph SR(n) (see Theorem 2.1 in the D. Antony Xavier et al. reference).

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.

Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

FORMULA

From Colin Barker, May 10 2018: (Start)

G.f.: x*(4 - 15*x + 15*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)

a(n) = A083313(n)+2. - R. J. Mathar, Jul 24 2022

MAPLE

seq(3^n-2^(n-1)+2, n = 1 .. 40);

PROG

(PARI) Vec(x*(4 - 15*x + 15*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, May 10 2018

(GAP) List([1..40], n->3^n-2^(n-1)+2); # Muniru A Asiru, May 10 2018