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A094855 Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n+1, s(0) = 4, s(2n+1) = 5. 3
1, 3, 10, 35, 124, 440, 1560, 5525, 19551, 69142, 244419, 863788, 3052100, 10782928, 38092457, 134560491, 475313762, 1678930611, 5930320300, 20946860064, 73987208296, 261331829501, 923052962407, 3260318517230, 11515766271219 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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+1)) counts (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < m and |s(i)-s(i-1)| = 1 for i = 1,2,....,2n+1, s(0) = j, s(2n+1) = k.

Counts all paths of length (2*n+1), n>=0, starting at the initial node on the path graph P_8, see the Maple program. - Johannes W. Meijer, May 29 2010

LINKS

Table of n, a(n) for n=0..24.

FORMULA

a(n)=(2/9)*Sum(r, 1, 8, Sin(4*r*Pi/9)Sin(5*r*Pi/9)(2Cos(r*Pi/9))^(2n+1)).

a(n)=7a(n-1)-15a(n-2)+10a(n-3)-a(n-4).

G.f.: (-1+2x)^2/(1-7x+15x^2-10x^3+x^4).

MAPLE

with(GraphTheory): G:=PathGraph(8): A:= AdjacencyMatrix(G): nmax:=24; for n from 0 to 2*nmax+2 do B(n):=A^n; a(n):=add(B(n)[1, k], k=1..8); od: seq(a(2*n+1), n=0..nmax); # Johannes W. Meijer, May 29 2010

CROSSREFS

a(n) = A061551(2*n+1). - Johannes W. Meijer, May 29 2010

Sequence in context: A159309 A112107 A187925 * A243871 A081567 A224509

Adjacent sequences:  A094852 A094853 A094854 * A094856 A094857 A094858

KEYWORD

nonn

AUTHOR

Herbert Kociemba, Jun 13 2004

STATUS

approved

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Last modified October 23 22:15 EDT 2019. Contains 328373 sequences. (Running on oeis4.)