A051436 Number of undirected walks of length n+1 on tetrahedron, visiting n+2 vertices, with n "corners", as in A001998, but allowing only rigid motions in 3-space (|G| = 12). Walks are not self-avoiding.

1, 2, 5, 12, 39, 111, 350, 1044, 3201, 9627, 29150, 87672, 264069, 793431, 2384450, 7159164, 21494001, 64507827, 193589270, 580878432, 1742897949, 5229157551, 15688522250, 47067483684, 141206647401, 423627793227, 1270900160990, 3812732430792, 11438264409429

Offset

0,2

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,0,-30,25,55,-60,-30,36).

Formula

n=2m: (3^n+3^m)/2 -2^(n-1)+2^(m-1); n=2m+1: (3^n+3^m)/2 - 2^(n-1) +1.

G.f.: -(39*x^7-20*x^6-39*x^5+14*x^4+17*x^3-5*x^2-3*x+1) / ((x-1)*(x+1)*(2*x-1)*(3*x-1)*(2*x^2-1)*(3*x^2-1)). - Colin Barker, Jul 17 2013

Example

For n=2 there are three walks that stay in one face and two that visit two faces.

Maple

a:= n-> `if`(irem(n, 2, 'm')=0,
        (3^n+3^m)/2+2^(m-1), (3^n+3^m)/2+1) -2^(n-1):
seq(a(n), n=0..35);  # Alois P. Heinz, Jul 17 2013

Mathematica

a[n_?OddQ] := (3^n + 3^((n - 1)/2))/2 - 2^(n - 1) + 1; a[n_?EvenQ] := (3^n + 3^(n/2))/2 - 2^(n - 1) + 2^(n/2 - 1); Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Jan 25 2013, from formula *)
LinearRecurrence[{5, 0, -30, 25, 55, -60, -30, 36}, {1, 2, 5, 12, 39, 111, 350, 1044}, 40] (* Harvey P. Dale, Oct 30 2015 *)

Prog

(Haskell)
a051436 n = (3 ^ n + 3 ^ m - 2 ^ n + (1 - r) * 2 ^ m) `div` 2 + r
            where (m, r) = divMod n 2
-- Reinhard Zumkeller, Jun 30 2013
(PARI) a(n)=if(n%2, (3^n + 3^((n - 1)/2))/2 + 1, (3^n + 3^(n/2))/2 + 2^(n/2 - 1)) - 2^(n-1) \\ Charles R Greathouse IV, Feb 10 2017

Crossrefs

Cf. A001998, A001444, A000079, A000244.

Sequence in context: A036782 A050237 A050258 * A054581 A203151 A140440

Adjacent sequences: A051433 A051434 A051435 * A051437 A051438 A051439

Keyword

nonn,walk,nice,easy

Author

Colin Mallows

Extensions

Corrected by T. D. Noe, Nov 09 2006

Status

approved