A302655 Number of minimal total dominating sets in the n-path graph.

0, 1, 2, 1, 2, 4, 3, 4, 8, 9, 10, 16, 21, 25, 36, 49, 60, 81, 112, 144, 189, 256, 336, 441, 592, 784, 1029, 1369, 1820, 2401, 3182, 4225, 5586, 7396, 9815, 12996, 17200, 22801, 30210, 40000, 53001, 70225, 93000, 123201, 163240, 216225, 286416, 379456, 502665

Offset

1,3

Links

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

Eric Weisstein's World of Mathematics, Path 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

From Andrew Howroyd, Apr 15 2018: (Start)

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*(1 + 2*x + x^2 + x^3 + x^4 - x^5 - 2*x^6 - x^7)/(1 - x^3 - x^4 - x^5 - x^6 + x^8 + x^9).

a(2*n) = A000931(n+5)^2. (End)

Mathematica

Table[If[Mod[n, 2] == 0, (RootSum[-1 - # + #^3 &, #^(n/2 + 5) (5 - 6 # + 4 #^2) &]/23)^2, (RootSum[-1 + # - 2 #^2 + #^3 &, #^((n - 1)/2) (4 - 2 # + 5 #^2) &] + RootSum[-1 + #^2 + #^3 &, #^((n - 1)/2) (-5 + 6 # + 3 #^2) &])/23], {n, 50}]
LinearRecurrence[{0, 0, 1, 1, 1, 1, 0, -1, -1}, {0, 1, 2, 1, 2, 4, 3, 4, 8}, 50]
CoefficientList[Series[(x (1 + 2 x + x^2 + x^3 + x^4 - x^5 - 2 x^6 - x^7))/(1 - x^3 - x^4 - x^5 - x^6 + x^8 + x^9), {x, 0, 50}], x]

Prog

(PARI) concat([0], Vec(x^2*(1 + 2*x + x^2 + x^3 + x^4 - x^5 - 2*x^6 - x^7)/(1 - x^3 - x^4 - x^5 - x^6 + x^8 + x^9) + O(x^50))) \\ Andrew Howroyd, Apr 15 2018

Crossrefs

Row 1 of A303118.

Cf. A000931, A300738, A302654.

Sequence in context: A238577 A131380 A100461 * A316997 A323465 A364780

Adjacent sequences: A302652 A302653 A302654 * A302656 A302657 A302658

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 11 2018

Extensions

Terms a(20) and beyond from Andrew Howroyd, Apr 15 2018

Status

approved