A303048 -id:A303048

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A303227

Number of minimal total dominating sets in the n-triangular (Johnson) graph.

+10
4

1, 1, 0, 3, 12, 80, 840, 4032, 31976, 371016, 4354650, 55066880, 680003412, 9047989392, 132626606294, 2096065474440, 34991505975120, 607163217989312, 11006996786618994, 209218563659672064, 4168806234781798100, 86745911047924139760, 1876774293382882814382

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..200

Eric Weisstein's World of Mathematics, Johnson Graph

Eric Weisstein's World of Mathematics, Total Dominating Set

Eric Weisstein's World of Mathematics, Triangular Graph

FORMULA

E.g.f.: exp(x^3/2) + x*exp(x*exp(x^2+x) - (x+x^2+x^3+x^4/2)). - Andrew Howroyd, Apr 21 2018

MATHEMATICA

Range[0, 20]! CoefficientList[Series[Exp[x^3/2] + x Exp[x Exp[x^2 + x] - (x + x^2 + x^3 + x^4/2)], {x, 0, 20}], x] (* Eric W. Weisstein, Apr 23 2018 *)

PROG

(PARI) seq(n)={Vec(serlaplace(exp(x^3/2 + O(x*x^n)) + x*exp(x*exp(x^2+x + O(x^n)) - (x+x^2+x^3+x^4/2))))} \\ Andrew Howroyd, Apr 21 2018

CROSSREFS

Cf. A290716, A303048, A304561.

KEYWORD

nonn

AUTHOR

Eric W. Weisstein, Apr 20 2018

EXTENSIONS

a(0)-a(1) prepended and a(8)-a(22) from Andrew Howroyd, Apr 21 2018

STATUS

approved

A304561

Number of minimum total dominating sets in the n-triangular (Johnson) graph.

+10
4

0, 3, 12, 80, 840, 630, 13440, 277200, 75600, 3326400, 116839800, 16216200, 1210809600, 65043178200, 5448643200, 617512896000, 47147109609600, 2639867630400, 422378820864000, 43505018548992000, 1742312636064000, 374016445875072000, 49991305310266320000, 1502744648605200000

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,2

COMMENTS

In general, a dominating set on a triangular graph corresponds with an edge cover on a complete graph with optionally one vertex uncovered. In the case of n mod 3 == 1, a minimum total dominating set will correspond with one uncovered vertex and the remaining covered by trees of size 3. In the case of n mod 3 == 2, one of trees needs to be increased to size 4. In the case of n divisible by 3, one tree may be size 5 or two size 4 or all may be size 3 but without an uncovered vertex. - Andrew Howroyd, May 20 2018

LINKS

Andrew Howroyd, Table of n, a(n) for n = 2..200

Eric Weisstein's World of Mathematics, Johnson Graph

Eric Weisstein's World of Mathematics, Total Dominating Set

Eric Weisstein's World of Mathematics, Triangular Graph

FORMULA

a(3*k+1) = (3*k+1)!/(2^k*k!), a(3*k+2) = 4*k*(3*k+2)!/(3*2^k*k!), a(3*k) = (18 - 11*k - 21*k^2 + 32*k^3)*(3*k)!/(18*2^k*k!). - Andrew Howroyd, May 20 2018

MATHEMATICA

Table[Piecewise[{{(2^-(n/3 + 1) (486 - 99 n - 63 n^2 + 32 n^3) n!)/(243 (n/3)!), Mod[n, 3] == 0}, {(2^((1 - n)/3) n!)/Gamma[(n + 2)/3], Mod[n, 3] == 1}, {(2^((8 - n)/3) n!)/(3 Gamma[(n - 2)/3]), Mod[n, 3] == 2}}], {n, 2, 30}]

PROG

(PARI) a(n)={my(t=n\3); n!*if(n%3==0, (18-11*t-21*t^2+32*t^3)/18, if(n%3==1, 1, 4*t/3))/(t!*(2^t))} \\ Andrew Howroyd, May 20 2018

CROSSREFS

Cf. A303048, A303227.

KEYWORD

nonn

AUTHOR

Eric W. Weisstein, May 14 2018

EXTENSIONS

a(9)-a(25) from Andrew Howroyd, May 20 2018

STATUS

approved

A323499

Number of minimum dominating sets in the n-triangular (Johnson) graph.

+10
1

1, 3, 15, 15, 195, 105, 2625, 945, 38745, 10395, 634095, 135135, 11486475, 2027025, 229053825, 34459425, 4996616625, 654729075, 118505962575, 13749310575, 3038597637075, 316234143225, 83802047954625, 7905853580625, 2474532170735625, 213458046676875

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,2

LINKS

Andrew Howroyd, Table of n, a(n) for n = 2..200

Eric Weisstein's World of Mathematics, Johnson Graph

Eric Weisstein's World of Mathematics, Minimum Dominating Set

Eric Weisstein's World of Mathematics, Triangular Graph

FORMULA

a(n) = n!! for n odd.

a(n) = (n-1)!!*(1 + n*(n/2 - 1)) for n even. - Andrew Howroyd, Sep 08 2019

PROG

(PARI) a(n)={my(m=(n+1)\2); ((2*m)!/(m!*2^m))*if(n%2, 1, 1 + n*(n/2-1))} \\ Andrew Howroyd, Sep 08 2019

CROSSREFS

Cf. A290716, A290847, A298104, A303048, A303227, A304561.

KEYWORD

nonn

AUTHOR

Eric W. Weisstein, Jan 16 2019

EXTENSIONS

Terms a(14) and beyond from Andrew Howroyd, Sep 08 2019

STATUS

approved

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