A284699 Number of dominating sets in the n-antiprism graph.

3, 15, 57, 223, 863, 3333, 12883, 49791, 192441, 743775, 2874655, 11110405, 42941187, 165965647, 641449337, 2479171199, 9581878847, 37033506309, 143132741651, 553201243263, 2138096511097, 8263641389887, 31938581194175, 123441098248197, 477093977471363, 1843945546253839, 7126761892007865

Offset: 1

Author

Eric W. Weisstein, Apr 01 2017

Keywords

• nonn
• easy

Comments

Recurrence used to extrapolate sequence to a(1) and a(2).

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Antiprism Graph
• Eric Weisstein's World of Mathematics, Dominating Set
• Index entries for linear recurrences with constant coefficients, signature (3,3,1,1,1).

Programs

Mathematica

LinearRecurrence[{3, 3, 1, 1, 1}, {3, 15, 57, 223, 863, 3333, 12883}, 20]
Table[RootSum[-1 - # - #^2 - 3 #^3 - 3 #^4 + #^5 &, #^n &], {n, 20}]

PARI

Vec(x*(3 + 6*x + 3*x^2 + 4*x^3 + 5*x^4) / (1 - 3*x - 3*x^2 - x^3 - x^4 - x^5) + O(x^30)) \\ Colin Barker, Apr 01 2017

Formula

From Colin Barker, Apr 01 2017: (Start)

G.f.: x*(3 + 6*x + 3*x^2 + 4*x^3 + 5*x^4) / (1 - 3*x - 3*x^2 - x^3 - x^4 - x^5).

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

(End)