A286985 Number of connected dominating sets in the n-prism graph.

7, 7, 39, 115, 343, 967, 2663, 7203, 19239, 50887, 133543, 348179, 902775, 2329607, 5986535, 15327555, 39115847, 99532423, 252601127, 639548595, 1615746455, 4073951559, 10253517671, 25763632995, 64635943783, 161928486727, 405134009511, 1012371656275

Offset: 1

Author

Eric W. Weisstein, May 17 2017

Keywords

• nonn

Comments

Sequence extrapolated to a(1) and a(2) using recurrence. - Andrew Howroyd, Sep 04 2017

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Connected Dominating Set
• Eric Weisstein's World of Mathematics, Prism Graph
• Index entries for linear recurrences with constant coefficients, signature (6, -11, 4, 5, -2, -1).

Crossrefs

Cf. A284702, A286182.

Programs

Mathematica

Rest @ CoefficientList[Series[x (7 - 35 x + 74 x^2 - 70 x^3 + 19 x^4 - 3 x^5)/((1 - x)^2*(1 - 2 x - x^2)^2), {x, 0, 28}], x] (* Michael De Vlieger, Sep 04 2017 *)
Table[LucasL[n, 2] + 2 n (3 Fibonacci[n - 2, 2] + Fibonacci[n - 1, 2] - 1) + 1, {n, 20}] (* Eric W. Weisstein, Sep 08 2017 *)
LinearRecurrence[{6, -11, 4, 5, -2, -1}, {7, 7, 39, 115, 343, 967}, 20] (* Eric W. Weisstein, Sep 08 2017 *)

PARI

Vec((7 - 35*x + 74*x^2 - 70*x^3 + 19*x^4 - 3*x^5)/((1 - x)^2*(1 - 2*x - x^2)^2) + O(x^30))

Formula

From Andrew Howroyd, Sep 04 2017: (Start)

a(n) = 6*a(n-1) - 11*a(n-2) + 4*a(n-3) + 5*a(n-4) - 2*a(n-5) - a(n-6) for n > 6.

G.f.: x*(7 - 35*x + 74*x^2 - 70*x^3 + 19*x^4 - 3*x^5)/((1 - x)^2*(1 - 2*x - x^2)^2).

(End)

Extensions

a(1)-a(2) and terms a(14) and beyond from Andrew Howroyd, Sep 04 2017