A303227 -id:A303227 - cp's OEIS frontend

A304561 Number of minimum total dominating sets in the n-triangular (Johnson) graph.

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

Offset: 2

Eric W. Weisstein, May 14 2018

nonn

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

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 Total Dominating Set.
Eric Weisstein's World of Mathematics, Triangular Graph.

Cf. A303048, A303227.

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}]

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

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

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

A303048 Number of total dominating sets in the n-triangular (Johnson) graph.

Original entry on oeis.org

0, 4, 54, 918, 31232, 2059624, 266734812, 68574627036, 35160753222400, 36021330363615408, 73782362964470935112, 302225854825997535378632, 2475866675779140063716682240, 40564755806137338166417907530592, 1329227401912999475682655581004557840

Offset: 2

Eric W. Weisstein, Apr 17 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 2..50
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

Cf. A006129, A290847, A298104, A303227, A304561.

b[n_] := Sum[(-1)^(n - k) Binomial[n, k] 2^Binomial[k, 2], {k, 0, n}]
Table[Sum[(-1)^k Binomial[n, 2 k] (2 k)!/(2^k k!) (b[n - 2 k] + (n - 2 k) b[n - 2 k - 1]), {k, 0, Floor[n/2]}], {n, 2, 20}]

\\ here b(n) is A006129
b(n)=sum(k=0, n, (-1)^(n-k)*binomial(n, k)*2^binomial(k, 2));
a(n)=sum(k=0, n\2, (-1)^k*binomial(n,2*k)*(2*k)!/(2^k*k!)*(b(n-2*k) + (n-2*k)*b(n-2*k-1))); \\ Andrew Howroyd, Apr 20 2018

a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n,2*k)*(2*k-1)!!*A290847(n-2*k). - Andrew Howroyd, Apr 20 2018

a(8)-a(16) from Andrew Howroyd, Apr 20 2018

A323499 Number of minimum dominating sets in the n-triangular (Johnson) graph.

Original entry on oeis.org

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

Offset: 2

Eric W. Weisstein, Jan 16 2019

nonn

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

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

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

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

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

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

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A304561 Number of minimum total dominating sets in the n-triangular (Johnson) graph.

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A303048 Number of total dominating sets in the n-triangular (Johnson) graph.

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A323499 Number of minimum dominating sets in the n-triangular (Johnson) graph.

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