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%I #25 May 14 2019 10:55:13
%S 6,36,384,5460,84816,1347396,21521184,344194740,5506552176,
%T 88102619556,1409633169984,22554096102420,360865400232336,
%U 5773845857280516,92381531540306784,1478104495968880500,23649671900884069296,378394750275931314276,6054316003862820691584
%N Number of directed Hamiltonian paths in the n-Hanoi graph.
%H Andrew Howroyd, <a href="/A137889/b137889.txt">Table of n, a(n) for n = 1..100</a>
%H Andrew Howroyd, <a href="/A137889/a137889.txt">Hamiltonian paths in Hanoi graphs</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianPath.html">Hamiltonian Path</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HanoiGraph.html">Hanoi Graph</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (24,-147,316,-192).
%F a(n) = (208 + 16*3^(n + 2) + 13*4^(n + 2) + 25*16^n)/312. - _Eric W. Weisstein_, Jun 19 2017
%F a(n) = 3*a(n-1) + (25*16^n + 64*4^n - 512)/384 for n > 1. - _Andrew Howroyd_, Jun 18 2017
%F From _Colin Barker_, Jul 30 2017: (Start)
%F G.f.: 6*x*(1 - 18*x + 67*x^2 - 60*x^3) / ((1 - x)*(1 - 3*x)*(1 - 4*x)*(1 - 16*x)).
%F a(n) = 24*a(n-1) - 147*a(n-2) + 316*a(n-3) - 192*a(n-4) for n>4.
%F (End)
%t Table[(208 + 16 3^(n + 2) + 13 4^(n + 2) + 25 16^n)/312, {n, 10}] (* _Eric W. Weisstein_, Jun 19 2017 *)
%t RecurrenceTable[{a[1] == 6, a[n] == 3 a[n - 1] + (25 16^n + 64 4^n - 512)/384}, a, {n, 10}] (* _Eric W. Weisstein_, Jun 19 2017 *)
%o (PARI) a(n)=if(n==1,6,3*a(n-1) + (25*16^n + 64*4^n - 512)/384); \\ _Andrew Howroyd_, Jun 18 2017
%o (PARI) Vec(6*x*(1 - 18*x + 67*x^2 - 60*x^3) / ((1 - x)*(1 - 3*x)*(1 - 4*x)*(1 - 16*x)) + O(x^30)) \\ _Colin Barker_, Jul 30 2017
%Y Cf. A288839 (chromatic polynomials of the n-Hanoi graph).
%Y Cf. A193233 (chromatic polynomial with highest coefficients first).
%Y Cf. A288490 (independent vertex sets in the n-Hanoi graph).
%Y Cf. A286017 (matchings in the n-Hanoi graph).
%Y Cf. A193136 (spanning trees of the n-Hanoi graph).
%Y Cf. A288796 (undirected paths in the n-Hanoi graph).
%K nonn,easy
%O 1,1
%A _Eric W. Weisstein_, Feb 20 2008
%E Terms a(5) and beyond from _Andrew Howroyd_, Jun 18 2017
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