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A302654 Number of minimum total dominating sets in the n-path graph. 3
0, 1, 2, 1, 1, 4, 3, 1, 2, 9, 4, 1, 3, 16, 5, 1, 4, 25, 6, 1, 5, 36, 7, 1, 6, 49, 8, 1, 7, 64, 9, 1, 8, 81, 10, 1, 9, 100, 11, 1, 10, 121, 12, 1, 11, 144, 13, 1, 12, 169, 14, 1, 13, 196, 15, 1, 14, 225, 16, 1, 15, 256, 17, 1, 16, 289, 18, 1, 17, 324, 19, 1, 18, 361, 20, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

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, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1).

FORMULA

a(n) = ((-1)^n*(n - 2)^2 + (6 + n)^2 - 2*(n - 2)*(n + 6)*cos(n*Pi/2) - 48*sin(n*Pi/2))/6.

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

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

MATHEMATICA

Table[Piecewise[{{1, Mod[n, 4] == 0}, {((n + 2)/4)^2, Mod[n, 4] == 2}, {(n - 1)/4, Mod[n, 4] == 1}, {(n + 5)/4, Mod[n, 4] == 3}}], {n, 20}]

Table[((-1)^n (n - 2)^2 + (6 + n)^2 - 2 (n - 2) (n + 6) Cos[n Pi/2] - 48 Sin[n Pi/2])/64, {n, 20}]

LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {0, 1, 2, 1, 1, 4, 3, 1, 2, 9, 4, 1}, 20]

PROG

(PARI) concat(0, Vec(x^2*(1 + 2*x + x^2 + x^3 + x^4 - 3*x^5 - 2*x^6 - x^7 + x^9 + x^10) / ((1 - x)^3*(1 + x)^3*(1 + x^2)^3) + O(x^70))) \\ Colin Barker, Dec 25 2019

CROSSREFS

Sequence in context: A125790 A294082 A129705 * A264831 A264728 A306790

Adjacent sequences:  A302651 A302652 A302653 * A302655 A302656 A302657

KEYWORD

nonn,easy

AUTHOR

Eric W. Weisstein, Apr 11 2018

STATUS

approved

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Last modified June 17 17:05 EDT 2021. Contains 345085 sequences. (Running on oeis4.)