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A300738 Number of minimal total dominating sets in the n-cycle graph. 4
0, 0, 3, 4, 5, 9, 7, 4, 12, 25, 22, 25, 39, 49, 68, 100, 119, 144, 209, 289, 367, 484, 644, 841, 1130, 1521, 1983, 2601, 3480, 4624, 6107, 8100, 10717, 14161, 18807, 24964, 33004, 43681, 57918, 76729, 101639, 134689, 178364, 236196, 313007, 414736, 549289 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Cycle Graph

Eric Weisstein's World of Mathematics, Total Dominating Set

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

FORMULA

a(n) = a(n-3) + a(n-4) + a(n-5) + a(n-6) - a(n-8) - a(n-9) for n > 9.

G.f.: x^2*(3 + 4*x + 5*x^2 + 6*x^3 - 8*x^5 - 9*x^6)/((1 - x^2 - x^3)*(1 + x^2 - x^6)).

a(2*n) = A001608(n)^2.

a(2*n-1) = A001608(2*n-1), where A001608 are the Perrin numbers.

MATHEMATICA

Table[RootSum[-1 - # + #^3 &, #^n &] + (1 + (-1)^n) RootSum[-1 + #^2 + #^3 &, #^(n/2) &], {n, 20}]

Perrin[n_] := RootSum[-1 - # + #^3 &, #^n &]; Table[With[{b = Mod[n, 2, 1]}, Perrin[n/b]^b], {n, 20}]

LinearRecurrence[{0, 0, 1, 1, 1, 1, 0, -1, -1}, {0, 0, 3, 4, 5, 9, 7, 4, 12}, 20]

CoefficientList[Series[x^2 (3 + 4 x + 5 x^2 + 6 x^3 - 8 x^5 - 9 x^6)/(1 - x^3 - x^4 - x^5 - x^6 + x^8 + x^9), {x, 0, 20}], x]

PROG

(PARI) concat([0, 0], Vec((3 + 4*x + 5*x^2 + 6*x^3 - 8*x^5 - 9*x^6)/((1 - x^2 - x^3)*(1 + x^2 - x^6)) + O(x^50)))

CROSSREFS

Cf. A001608, A001638 (total dominating sets), A253413, A302653, A302655, A302918.

Sequence in context: A122401 A122403 A302653 * A324506 A065336 A079097

Adjacent sequences:  A300735 A300736 A300737 * A300739 A300740 A300741

KEYWORD

nonn

AUTHOR

Andrew Howroyd, Apr 15 2018

STATUS

approved

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Last modified December 12 15:11 EST 2019. Contains 329960 sequences. (Running on oeis4.)