A302653 Number of minimum total dominating sets in the n-cycle graph.

1, 1, 3, 4, 5, 9, 7, 4, 9, 25, 11, 4, 13, 49, 15, 4, 17, 81, 19, 4, 21, 121, 23, 4, 25, 169, 27, 4, 29, 225, 31, 4, 33, 289, 35, 4, 37, 361, 39, 4, 41, 441, 43, 4, 45, 529, 47, 4, 49, 625, 51, 4, 53, 729, 55, 4, 57, 841, 59, 4, 61, 961, 63, 4, 65, 1089, 67, 4, 69, 1225, 71, 4

Offset

1,3

Comments

Extended to a(1)-a(2) using the formula/recurrence.

Links

Colin Barker, 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 (0,0,0,3,0,0,0,-3,0,0,0,1).

Formula

a(n) = n for odd n.

a(n) = 4 for n mod 4 = 0.

a(n) = (n/2)^2 for n mod 4 = 2.

a(n) = ((-1)^n*(n - 4)^2 + (n + 4)^2 - 2*(n - 4)*(n + 4)*cos(n*Pi/2))/16.

a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).

G.f.: x*(1 + x + 3*x^2 + 4*x^3 + 2*x^4 + 6*x^5 - 2*x^6 - 8*x^7 - 3*x^8 + x^9 - x^10 + 4*x^11) / ((1 - x)^3*(1 + x)^3*(1 + x^2)^3). - Colin Barker, Dec 25 2019

Mathematica

Table[((-1)^n (n - 4)^2 + (n + 4)^2 - 2 (n - 4) (n + 4) cos(n Pi/2))/16, {n, 80}]
Table[Piecewise[{{n, Mod[n, 2] == 1}, {4, Mod[n, 4] == 0}, {(n/2)^2, Mod[n, 4] == 2}}], {n, 80}]
LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {1, 1, 3, 4, 5, 9, 7, 4, 9, 25, 11, 4}, 80]

Prog

(PARI) Vec(x*(1 + x + 3*x^2 + 4*x^3 + 2*x^4 + 6*x^5 - 2*x^6 - 8*x^7 - 3*x^8 + x^9 - x^10 + 4*x^11) / ((1 - x)^3*(1 + x)^3*(1 + x^2)^3) + O(x^70)) \\ Colin Barker, Dec 25 2019

Crossrefs

Sequence in context: A122401 A122403 A349659 * A300738 A324506 A342772

Adjacent sequences: A302650 A302651 A302652 * A302654 A302655 A302656

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 11 2018

Status

approved