A303226 Number of minimal total dominating sets in the n-gear graph.

0, 6, 12, 6, 30, 30, 56, 110, 156, 306, 506, 870, 1560, 2652, 4692, 8190, 14280, 25122, 43890, 77006, 135056, 236682, 415380, 728462, 1278030, 2242506, 3934272, 6903756, 12113880, 21256710, 37301556, 65456190, 114864806, 201569006, 353722056, 620732310

Offset

1,2

Comments

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, Apr 20 2018

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Eric Weisstein's World of Mathematics, Gear Graph

Eric Weisstein's World of Mathematics, Total Dominating Set

Index entries for linear recurrences with constant coefficients, signature (1,2,1,-3,-1,-1,0,0,1).

Formula

From Andrew Howroyd, Apr 20 2018: (Start)

a(n) = a(n-1) + 2*a(n-2) + a(n-3) - 3*a(n-4) - a(n-5) - a(n-6) + a(n-9) for n > 9.

G.f.: 2*x^2*(3 + 3*x - 9*x^2 - 3*x^3 - 3*x^4 + x^5 + 6*x^7)/((1 - 2*x + x^2 - x^3)*(1 + x - x^3)*(1 - x^2 - x^3)).

(End)

Mathematica

Table[RootSum[-1 - # + #^3 &, #^n &] + RootSum[-1 + # - 2 #^2 + #^3 &, #^n &] + 2 RootSum[-1 + #^2 + #^3 &, #^(n + 2) (1 + #) &], {n, 20}]
LinearRecurrence[{1, 2, 1, -3, -1, -1, 0, 0, 1}, {0, 6, 12, 6, 30, 30, 56, 110, 156}, 20]
CoefficientList[Series[-2 x (3 + 3 x - 9 x^2 - 3 x^3 - 3 x^4 + x^5 + 6 x^7)/(-1 + x + 2 x^2 + x^3 - 3 x^4 - x^5 - x^6 + x^9), {x, 0, 20}], x]

Prog

(PARI) concat([0], Vec(2*(3 + 3*x - 9*x^2 - 3*x^3 - 3*x^4 + x^5 + 6*x^7)/((1 - 2*x + x^2 - x^3)*(1 + x - x^3)*(1 - x^2 - x^3)) + O(x^40))) \\ Andrew Howroyd, Apr 20 2018

Crossrefs

Cf. A290378, A290938.

Sequence in context: A343052 A050496 A262617 * A360877 A295122 A103698

Adjacent sequences: A303223 A303224 A303225 * A303227 A303228 A303229

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 20 2018

Extensions

a(1)-a(2) and terms a(11) and beyond from Andrew Howroyd, Apr 20 2018

Status

approved