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%I #17 Jan 04 2024 18:10:09
%S 0,3,12,80,840,630,13440,277200,75600,3326400,116839800,16216200,
%T 1210809600,65043178200,5448643200,617512896000,47147109609600,
%U 2639867630400,422378820864000,43505018548992000,1742312636064000,374016445875072000,49991305310266320000,1502744648605200000
%N Number of minimum total dominating sets in the n-triangular (Johnson) graph.
%C 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
%H Andrew Howroyd, <a href="/A304561/b304561.txt">Table of n, a(n) for n = 2..200</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JohnsonGraph.html">Johnson Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotalDominatingSet.html">Total Dominating Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>
%F 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
%t 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}]
%o (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
%Y Cf. A303048, A303227.
%K nonn
%O 2,2
%A _Eric W. Weisstein_, May 14 2018
%E a(9)-a(25) from _Andrew Howroyd_, May 20 2018
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