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A303212 Number of minimum total dominating sets in the n X n rook complement graph. 2
0, 1, 6, 96, 600, 2400, 7350, 18816, 42336, 86400, 163350, 290400, 490776, 794976, 1242150, 1881600, 2774400, 3995136, 5633766, 7797600, 10613400, 14229600, 18818646, 24579456, 31740000, 40560000, 51333750, 64393056, 80110296, 98901600 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

For n > 2, the minimum total dominating sets are any three vertices such that no two are in the same row or column. - Andrew Howroyd, Apr 20 2018

LINKS

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

Eric Weisstein's World of Mathematics, Rook Complement Graph

Eric Weisstein's World of Mathematics, Total Dominating Set

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

FORMULA

a(n) = A179058(n) for n > 2. - Andrew Howroyd, Apr 20 2018

From Colin Barker, Apr 20 2018: (Start)

G.f.: x^2*(1 - x + 75*x^2 + 19*x^3 + 41*x^4 - 21*x^5 + 7*x^6 - x^7) / (1 - x)^7.

a(n) = n^2*(2 - 3*n + n^2)^2 / 6 for n>2.

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>9.

(End)

MATHEMATICA

Table[If[n == 2, 1, 6 Binomial[n, 3]^2], {n, 20}]

Join[{0, 1}, LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 6, 96, 600, 2400, 7350}, {3, 20}]]

CoefficientList[Series[x (-1 + x - 75 x^2 - 19 x^3 - 41 x^4 + 21 x^5 - 7 x^6 + x^7)/(-1 + x)^7, {x, 0, 20}], x]

PROG

(PARI) a(n) = if(n<3, n==2, 6*binomial(n, 3)^2) \\ Andrew Howroyd, Apr 20 2018

(PARI) concat(0, Vec(x^2*(1 - x + 75*x^2 + 19*x^3 + 41*x^4 - 21*x^5 + 7*x^6 - x^7) / (1 - x)^7 + O(x^60))) \\ Colin Barker, Apr 20 2018

CROSSREFS

Cf. A179058.

Sequence in context: A222971 A196813 A179058 * A226549 A053338 A115400

Adjacent sequences:  A303209 A303210 A303211 * A303213 A303214 A303215

KEYWORD

nonn,easy

AUTHOR

Eric W. Weisstein, Apr 19 2018

EXTENSIONS

a(6)-a(30) from Andrew Howroyd, Apr 20 2018

STATUS

approved

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Last modified September 17 22:53 EDT 2019. Contains 327147 sequences. (Running on oeis4.)