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

Author

Eric W. Weisstein, Apr 20 2018

Keywords

• nonn
• easy

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, Minimal Total Dominating Set.
• Index entries for linear recurrences with constant coefficients, signature (1,2,1,-3,-1,-1,0,0,1).

Crossrefs

Cf. A141310, A290378, A290938, A303834.

Programs

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]

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

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)

Extensions

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