A094854 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 4, s(2n) = 4.

1, 2, 6, 20, 69, 241, 846, 2977, 10490, 36994, 130532, 460737, 1626629, 5743674, 20283121, 71632290, 252989326, 893528468, 3155899165, 11146628105, 39370204614, 139057473905, 491159630010, 1734810719530, 6127485120996

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) 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.

Counts all paths of length (2*n), 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..1825
• Index entries for linear recurrences with constant coefficients, signature (7,-15,10,-1).

Crossrefs

Even bisection of A061551.

Cf. A094855.

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),n=0..nmax); # Johannes W. Meijer, May 29 2010

Mathematica

CoefficientList[Series[-(2 x - 1) (x^2 - 3 x + 1)/((x - 1) (x^3 - 9 x^2 + 6 x - 1)), {x, 0, 24}], x] (* Michael De Vlieger, Feb 12 2022 *)

Formula

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

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

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

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