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A303054 Number of minimum total dominating sets in the n-ladder graph. 3
1, 4, 1, 16, 9, 1, 64, 16, 1, 169, 25, 1, 361, 36, 1, 676, 49, 1, 1156, 64, 1, 1849, 81, 1, 2809, 100, 1, 4096, 121, 1, 5776, 144, 1, 7921, 169, 1, 10609, 196, 1, 13924, 225, 1, 17956, 256, 1, 22801, 289, 1, 28561, 324, 1, 35344, 361, 1, 43264, 400, 1, 52441 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Each vertex can dominate up to three others. A ladder with a length that is an exact multiple of three can be dominated in only one way with 2n/3 vertices. - Andrew Howroyd, Apr 21 2018

LINKS

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

Eric Weisstein's World of Mathematics, Ladder Graph

Eric Weisstein's World of Mathematics, Total Dominating Set

Index entries for linear recurrences with constant coefficients, signature (0,0,5,0,0,-10,0,0,10,0,0,-5,0,0,1).

FORMULA

a(n) = 1 for n mod 3 = 0

     = ((n^2 + 13*n + 4)/18)^2 for n mod 3 = 1

     = ((n + 4)/3)^2 for n mod 3 = 2.

G.f.: x*(-1 - 4*x - x^2 - 11*x^3 + 11*x^4 + 4*x^5 + 6*x^6 - 11*x^7 - 6*x^8 + x^9 + 5*x^10 + 4*x^11 - x^12 - x^13 - x^14)/(-1 + x^3)^5.

a(n) = 5*a(n-3) - 10*a(n-6) + 10*a(n-9) - 5*a(n-12) + a(n-15) for n>15. - Colin Barker, Apr 23 2018

EXAMPLE

From Andrew Howroyd, Apr 21 2018: (Start)

a(9) = 1 because there is only one arrangement of 6 vertices that is totally dominating and no set with fewer vertices can be totally dominating:

  .__o__.__.__o__.__.__o__.

     |        |        |

  .__o__.__.__o__.__.__o__.

(End)

MATHEMATICA

Table[Piecewise[{{1, Mod[n, 3] == 0}, {((n^2 + 13 n + 4)/18)^2, Mod[n, 3] == 1}, {((n + 4)/3)^2, Mod[n, 3] == 2}}], {n, 58}] (* Eric W. Weisstein, Apr 23 2018 and Michael De Vlieger, Apr 21 2018 *)

Table[(916 + 392 n + 213 n^2 + 26 n^3 + n^4 - (-56 + 392 n + 213 n^2 + 26 n^3 + n^4) Cos[2 n Pi/3] + Sqrt[3] (-20 + 7 n + n^2) (28 + 19 n + n^2) Sin[2 n Pi/3])/972, {n, 20}] (* Eric W. Weisstein, Apr 23 2018 *)

LinearRecurrence[{0, 0, 5, 0, 0, -10, 0, 0, 10, 0, 0, -5, 0, 0, 1}, {1, 4, 1, 16, 9, 1, 64, 16, 1, 169, 25, 1, 361, 36, 1}, 20] (* Eric W. Weisstein, Apr 23 2018 *)

CoefficientList[Series[(-1 - 4 x - x^2 - 11 x^3 + 11 x^4 + 4 x^5 + 6 x^6 - 11 x^7 - 6 x^8 + x^9 + 5 x^10 + 4 x^11 - x^12 - x^13 - x^14)/(-1 + x^3)^5, {x, 0, 20}], x] (* Eric W. Weisstein, Apr 23 2018 *)

PROG

(PARI) a(n)={if(n%3==0, 1, if(n%3==1, (n^2 + 13*n + 4)/18, (n + 4)/3))^2} \\ Andrew Howroyd, Apr 21 2018

(PARI) Vec(x*(1 + 4*x + x^2 + 11*x^3 - 11*x^4 - 4*x^5 - 6*x^6 + 11*x^7 + 6*x^8 - x^9 - 5*x^10 - 4*x^11 + x^12 + x^13 + x^14) / ((1 - x)^5*(1 + x + x^2)^5) + O(x^60)) \\ Colin Barker, Apr 23 2018

CROSSREFS

Row 2 of A303293.

Sequence in context: A138681 A038231 A104855 * A143496 A143697 A272088

Adjacent sequences:  A303051 A303052 A303053 * A303055 A303056 A303057

KEYWORD

nonn,easy

AUTHOR

Eric W. Weisstein, Apr 17 2018

EXTENSIONS

Terms a(14) and beyond from Andrew Howroyd, Apr 21 2018

STATUS

approved

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Last modified September 20 20:02 EDT 2020. Contains 337265 sequences. (Running on oeis4.)