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A052975 Expansion of (1-2*x)*(1-x)/(1-5*x+6*x^2-x^3). 11
1, 2, 6, 19, 61, 197, 638, 2069, 6714, 21794, 70755, 229725, 745889, 2421850, 7863641, 25532994, 82904974, 269190547, 874055885, 2838041117, 9215060822, 29921113293, 97153242650, 315454594314, 1024274628963, 3325798821581, 10798800928441, 35063486341682 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 7 and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n, s(0) = 3, s(2n) = 3. - Herbert Kociemba, Jun 11 2004

Counts all paths of length (2*n), n>=0, starting at the initial node and ending on the nodes 1, 2, 3, 4 and 5 on the path graph P_6, see the second Maple program. - Johannes W. Meijer, May 29 2010

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1047

Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.

Index entries for linear recurrences with constant coefficients, signature (5,-6,1).

FORMULA

G.f.: (1-2*x)*(1-x)/(1-5*x+6*x^2-x^3).

a(n) = A028495(2*n). - Floor van Lamoen, Nov 02 2005

a(n) = Sum (1/7*(2-3*_alpha+_alpha^2)*_alpha^(-1-n), _alpha=RootOf(-1+5*_Z-6*_Z^2+_Z^3))

From Herbert Kociemba, Jun 11 2004: (Start)

a(n) = (2/7)*Sum(r=1..6, sin(r*3*Pi/7)^2*(2*cos(r*Pi/7))^(2*n)).

a(n) = 5*a(n-1) -6*a(n-2) +a(n-3). (End)

a(n) = 2^n*A(n;1/2)=(1/7)*(s(2)^2*c(4)^(2n) + s(4)^2*c(1)^(2n) + s(1)^2*c(2)^(2n)), where c(j):=2*cos(2Pi*j/7) and s(j):=2*Sin(2Pi*j/7). Here A(n;d), n in N, d in C denotes the respective quasi-Fibonacci number - see A121449 and Witula-Slota-Warzynski paper for details (see also A094789, A085810, A077998, A006054, A121442). I note that my and the respective Herbert Kociemba's formulas are "compatible". - Roman Witula, Aug 09 2012

a(n) = A005021(n)-3*A005021(n-1)+2*A005021(n-2). - R. J. Mathar, Feb 27 2019

MAPLE

spec := [S, {S=Sequence(Prod(Union(Sequence(Prod(Sequence(Z), Z)), Sequence(Z)), Z))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);

with(GraphTheory):G:=PathGraph(6): A:= AdjacencyMatrix(G): nmax:=25; n2:=2*nmax+1: for n from 0 to n2 do B(n):=A^n; a(n):=add(B(n)[k, 1], k=1..5); od: seq(a(2*n), n=0..nmax); # Johannes W. Meijer, May 29 2010

MATHEMATICA

LinearRecurrence[{5, -6, 1}, {1, 2, 6}, 50] (* Roman Witula, Aug 09 2012 *)

CoefficientList[Series[(1 - 2 x) (1 - x)/(1 - 5 x + 6 x^2 - x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 18 2015 *)

PROG

(MAGMA) I:=[1, 2, 6]; [n le 3 select I[n] else 5*Self(n-1)-6*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Sep 18 2015

(PARI) x='x+O('x^30); Vec((1-2*x)*(1-x)/(1-5*x+6*x^2-x^3)) \\ G. C. Greubel, Apr 19 2018

CROSSREFS

Cf. A060557.

Cf. A028495, A078038 and A094790.

Sequence in context: A014010 A022015 A138747 * A275943 A228180 A035929

Adjacent sequences:  A052972 A052973 A052974 * A052976 A052977 A052978

KEYWORD

easy,nonn

AUTHOR

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

STATUS

approved

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Last modified August 24 16:16 EDT 2019. Contains 326295 sequences. (Running on oeis4.)