A137889 Number of directed Hamiltonian paths in the n-Hanoi graph.

6, 36, 384, 5460, 84816, 1347396, 21521184, 344194740, 5506552176, 88102619556, 1409633169984, 22554096102420, 360865400232336, 5773845857280516, 92381531540306784, 1478104495968880500, 23649671900884069296, 378394750275931314276, 6054316003862820691584

Offset: 1

Author

Eric W. Weisstein, Feb 20 2008

Keywords

• nonn
• easy

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..100
• Andrew Howroyd, Hamiltonian paths in Hanoi graphs
• Eric Weisstein's World of Mathematics, Hamiltonian Path
• Eric Weisstein's World of Mathematics, Hanoi Graph
• Index entries for linear recurrences with constant coefficients, signature (24,-147,316,-192).

Crossrefs

Cf. A288839 (chromatic polynomials of the n-Hanoi graph).

Cf. A193233 (chromatic polynomial with highest coefficients first).

Cf. A288490 (independent vertex sets in the n-Hanoi graph).

Cf. A286017 (matchings in the n-Hanoi graph).

Cf. A193136 (spanning trees of the n-Hanoi graph).

Cf. A288796 (undirected paths in the n-Hanoi graph).

Programs

Mathematica

Table[(208 + 16 3^(n + 2) + 13 4^(n + 2) + 25 16^n)/312, {n, 10}] (* Eric W. Weisstein, Jun 19 2017 *)
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 *)

PARI

a(n)=if(n==1,6,3*a(n-1) + (25*16^n + 64*4^n - 512)/384); \\ Andrew Howroyd, Jun 18 2017

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

Formula

a(n) = (208 + 16*3^(n + 2) + 13*4^(n + 2) + 25*16^n)/312. - Eric W. Weisstein, Jun 19 2017

a(n) = 3*a(n-1) + (25*16^n + 64*4^n - 512)/384 for n > 1. - Andrew Howroyd, Jun 18 2017

From Colin Barker, Jul 30 2017: (Start)

G.f.: 6*x*(1 - 18*x + 67*x^2 - 60*x^3) / ((1 - x)*(1 - 3*x)*(1 - 4*x)*(1 - 16*x)).

a(n) = 24*a(n-1) - 147*a(n-2) + 316*a(n-3) - 192*a(n-4) for n>4.

(End)

Extensions

Terms a(5) and beyond from Andrew Howroyd, Jun 18 2017