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A302653 Number of minimum total dominating sets in the n-cycle graph. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A215496 A122401 A122403 * A300738 A324506 A065336

Adjacent sequences:  A302650 A302651 A302652 * A302654 A302655 A302656

KEYWORD

nonn,easy

AUTHOR

Eric W. Weisstein, Apr 11 2018

STATUS

approved

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Last modified May 28 21:37 EDT 2020. Contains 334690 sequences. (Running on oeis4.)