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.

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

Offset: 0

Author

Herbert Kociemba, Jun 13 2004

Keywords

• nonn
• easy

Comments

In general, a(n) = (2/m)*Sum_{r=1..m-1} sin(r*j*Pi/m)*sin(r*k*Pi/m)*(2*cos(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

• Michael De Vlieger, Table of n, a(n) for n = 0..1824
• Index entries for linear recurrences with constant coefficients, signature (7,-15,10,-1).

Crossrefs

Odd bisection of A061551.

Cf. A094854.

Programs

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

Mathematica

LinearRecurrence[{7,-15,10,-1},{1,3,10,35},30] (* Harvey P. Dale, Jan 17 2022 *)

Formula

a(n) = (2/9)*Sum_{r=1..8} sin(4*r*Pi/9)*sin(5*r*Pi/9)*(2*cos(r*Pi/9))^(2n+1).

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

G.f.: (2*x-1)^2 / ( (x-1)*(x^3-9*x^2+6*x-1) ).

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