A054878 Number of closed walks of length n along the edges of a tetrahedron based at a vertex.

1, 0, 3, 6, 21, 60, 183, 546, 1641, 4920, 14763, 44286, 132861, 398580, 1195743, 3587226, 10761681, 32285040, 96855123, 290565366, 871696101, 2615088300, 7845264903, 23535794706, 70607384121, 211822152360, 635466457083

Offset: 0

Author

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

Keywords

• nonn
• walk
• easy

Comments

Number of closed walks of length n at a vertex of C_4, the cyclic graph on 4 nodes. 3*A015518(n) + a(n) = 3^n. - Paul Barry, Feb 03 2004

Form the digraph with matrix A = [0,1,1,1; 1,0,1,1; 1,1,0,1; 1,0,1,1]; a(n) corresponds to the (1,1) term of A^n. - Paul Barry, Oct 02 2004

Absolute values of A084567 (compare generating functions).

For n > 1, 4*a(n)=A218034(n)= the trace of the n-th power of the adjacency matrix for a complete 4-graph, a 4 X 4 matrix with a null diagonal and all ones for off-diagonal elements. The diagonal elements for the n-th power are a(n) and the off-diagonal are a(n)+1 for an odd power and a(n)-1 for an even (cf. A001045). - Tom Copeland, Nov 06 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
• M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
• R. J. Mathar, Counting Walks on Finite Graphs, (Nov 2020), Section 2.
• Index entries for linear recurrences with constant coefficients, signature (2,3).

Crossrefs

Row n=4 of A109502. A084567 (signed version).

{a(n)/3} for n>0 is A015518, non-closed walks.

Cf. A001045, A078008, A097073, A115341, A015518 (sequences where a(n)=3^n-a(n-1)). - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

Programs

Magma

[(3^n+(-1)^n*3)/4: n in [0..35]]; // Vincenzo Librandi, Jun 30 2011

Maple

A054878:=n->(3^n + (-1)^n*3)/4: seq(A054878(n), n=0..50); # Wesley Ivan Hurt, Sep 16 2017

Mathematica

Table[(3^n + (-1)^n*3)/4, {n, 0, 26}] (* or *)
CoefficientList[Series[1/4*(3/(1 + x) + 1/(1 - 3 x)), {x, 0, 26}], x] (* Michael De Vlieger, Sep 15 2017 *)

PARI

a(n) = (3^n + 3*(-1)^n)/4; \\ Altug Alkan, Sep 17 2017

Formula

a(n) = (3^n + (-1)^n*3)/4.

G.f.: 1/4*(3/(1+x) + 1/(1-3*x)).

E.g.f.: (exp(3*x) + 3*exp(-x))/4. - Paul Barry, Apr 20 2003

a(n) = 3^n - a(n-1) with a(0)=0. - Labos Elemer, Apr 26 2003

G.f.: (1 - 3*x^2 - 2*x^3)/(1 - 6*x^2 - 8*x^3 - 3*x^4) = (1 - 3*x^2 - 2*x^3)/charpoly(adj(C_4)). - Paul Barry, Feb 03 2004

From Paul Barry, Oct 02 2004: (Start)

G.f.: (1-2*x)/(1 - 2*x - 3*x^2).

a(n) = 2*a(n-1) + 3*a(n-2). (End)

G.f.: 1 - x + x/Q(0), where Q(k) = 1 + 3*x^2 - (3*k+4)*x + x*(3*k+1 - 3*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013

a(n+m) = a(n)*a(m) + a(n+1)*a(m+1)/3. - Yuchun Ji, Sep 12 2017

a(n) = 3*a(n-1) + 3*(-1)^n. - Yuchun Ji, Sep 13 2017

From Peter Bala, May 28 2024: (Start)

a(n) = (-1)^n + Sum_{k = 1..n} (-1)^(n-k)*binomial(n, k)*4^(k-1).

G.f.: A(x) = 1/(1 - x^2) o 1/(1 - x^2), where o denotes the black diamond product of power series as defined by Dukes and White. Cf. A015575.

The black diamond product A(x) o A(x) is the g.f. for the number of closed walks of length n at a vertex along the edges of the 15-simplex. (End)