A094833 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) = 3, s(2n) = 5.

1, 4, 15, 55, 199, 714, 2548, 9061, 32148, 113887, 403051, 1425471, 5039254, 17809336, 62928201, 222324436, 785402143, 2774421135, 9800231959, 34617003682, 122274355596, 431893332397, 1525507797700, 5388281150223

Offset: 1

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.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1825
• Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
• Index entries for linear recurrences with constant coefficients, signature (6,-9,1).

Programs

Mathematica

Rest@ CoefficientList[Series[(-x + 2 x^2)/(-1 + 6 x - 9 x^2 + x^3), {x, 0, 24}], x] (* Michael De Vlieger, Jul 02 2021 *)

Formula

a(n) = (2/9)*Sum_{r=1..8} sin(r*Pi/3)*sin(5*r*Pi/9)*(2*cos(r*Pi/9))^(2n).

a(n) = 6a(n-1) - 9a(n-2) + a(n-3).

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

a(n+1) = 3*a(n) + A094832(n-1). - Philippe Deléham, Mar 20 2007

a(n) = A094829(n+1) - 2*A094829(n). - R. J. Mathar, Nov 14 2019