Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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