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A054878 Number of closed walks of length n along the edges of a tetrahedron based at a vertex. 23
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

Index entries for linear recurrences with constant coefficients, signature (2,3).

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

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 *)

PROG

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

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

CROSSREFS

{a(n)/3} for n>0 is A015518.

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

Sequence in context: A148622 A148623 A259273 * A084567 A294527 A261582

Adjacent sequences:  A054875 A054876 A054877 * A054879 A054880 A054881

KEYWORD

nonn,walk,easy

AUTHOR

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

STATUS

approved

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Last modified November 21 07:11 EST 2017. Contains 294996 sequences.