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

Author

Eric W. Weisstein, Apr 11 2018

Keywords

• nonn
• easy

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Minimal Total Dominating Set.
• Eric Weisstein's World of Mathematics, Path Graph.
• Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,1,1,0,-1,-1).

Crossrefs

Row 1 of A303118.

Cf. A000931, A300738, A302654.

Programs

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]

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

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)

Extensions

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