A094817 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 3.

2, 6, 19, 62, 206, 692, 2340, 7944, 27032, 92112, 314128, 1071776, 3657824, 12485696, 42623040, 145512576, 496787840, 1696093440, 5790732544, 19770612224, 67500721664, 230461137920, 786842059776, 2686443866112

Offset: 1

Author

Herbert Kociemba, Jun 12 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.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1875
• Index entries for linear recurrences with constant coefficients, signature (6,-10,4).

Crossrefs

Cf. A007052, A216232.

Programs

Mathematica

Rest@ CoefficientList[Series[-x (2 - 6 x + 3 x^2)/((2 x - 1) (2 x^2 - 4 x + 1)), {x, 0, 24}], x] (* Michael De Vlieger, Feb 12 2022 *)

Formula

a(n) = (1/4) * Sum_{r=1..7} sin(3*r*Pi/8)^2*(2*cos(r*Pi/8))^(2*n).

a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3), n >= 4.

G.f.: -x*(2-6*x+3*x^2) / ( (2*x-1)*(2*x^2-4*x+1) ).

a(n) = A216232(n,n), for n >= 1. - Philippe Deléham, Mar 21 2013

4*a(n) = 2*A007052(n) + 2^n. - R. J. Mathar, Nov 14 2019