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A094790 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) = 1, s(2n) = 3. 13
1, 3, 9, 28, 89, 286, 924, 2993, 9707, 31501, 102256, 331981, 1077870, 3499720, 11363361, 36896355, 119801329, 388991876, 1263047761, 4101088878, 13316149700, 43237262993, 140390505643, 455845099957, 1480119728920 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

In general a(n)= 2/m*Sum_{r=1..m-1} sin(r*j*Pi/m)sin(r*k*Pi/m)(2cos(r*Pi/m))^(2n)) counts (s(0), s(1), ..., s(2n)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n, s(0) = j, s(2n) = k.

With interpolated zeros (0,0,1,0,3,0,9...), counts walks of length n between the first and third nodes of P_6. - Paul Barry, Jan 26 2005

Counts all paths of length (2*n+1), 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 Maple program. - Johannes W. Meijer, May 29 2010

With offset 0 = the INVERT transform of A055588. - Gary W. Adamson, Apr 01 2011

LINKS

Table of n, a(n) for n=1..25.

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

FORMULA

a(n)=2/7*Sum_{k=1..6} sin(Pi*k/7)sin(3Pi*k/7)(2cos(Pi*k/7))^(2n).

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

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

a(n) = rightmost term in M^n * [1,0,0] where M = the 3 X 3 matrix [2,1,1; 1,2,0; 1,0,1]. E.g., M^3 * [1,0,0] = [19,14,9]; right term = 9 = a(3). - Gary W. Adamson, Apr 04 2006

MAPLE

with(GraphTheory):G:=PathGraph(6): A:= AdjacencyMatrix(G): nmax:=24; 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+1), n=0..nmax); # Johannes W. Meijer, May 29 2010

MATHEMATICA

f[n_] := FullSimplify[ TrigToExp[(2/7)Sum[ Sin[Pi*k/7]Sin[3Pi*k/7](2Cos[Pi*k/7])^(2n), {k, 1, 6}]]]; Table[ f[n], {n, 25}] (* Robert G. Wilson v, Jun 18 2004 *)

PROG

(PARI) Vec(x*(-1+2*x)/(-1+5*x-6*x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Jun 14 2015

CROSSREFS

Cf. A080937, A094789, A005021, A028495, A052975, A078038, A055588.

Sequence in context: A170953 A199104 A049220 * A007822 A094164 A094803

Adjacent sequences:  A094787 A094788 A094789 * A094791 A094792 A094793

KEYWORD

nonn,easy

AUTHOR

Herbert Kociemba, Jun 11 2004

STATUS

approved

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Last modified August 20 01:14 EDT 2019. Contains 326136 sequences. (Running on oeis4.)