A094788 Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 1, s(2n+1) = 6.

1, 6, 27, 110, 428, 1624, 6069, 22458, 82555, 302082, 1101816, 4009616, 14567657, 52865230, 191684283, 694609494, 2515972324, 9110338728, 32981059485, 119377761602, 432046756571, 1563510554986, 5657752486512, 20472344560800

Offset: 2

Author

Herbert Kociemba, Jun 15 2004

Keywords

• nonn
• easy

Comments

Diagonal of the square array A217593. - Philippe Deléham, Mar 28 2013

Links

• Michael De Vlieger, Table of n, a(n) for n = 2..1791
• László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
• Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Programs

Mathematica

Drop[CoefficientList[Series[-x^2*(-1 + 2 x)/((x^2 - 3 x + 1) (5 x^2 - 5 x + 1)), {x, 0, 25}], x], 2] (* Michael De Vlieger, Aug 04 2021 *)
LinearRecurrence[{8,-21,20,-5},{1,6,27,110},30] (* Harvey P. Dale, Aug 31 2021 *)

PARI

Vec(x^2*(1-2*x)/(1-8*x+21*x^2-20*x^3+5*x^4)+O(x^66)) /* Joerg Arndt, Mar 29 2013 */

Formula

a(n) = (1/5)*Sum_{r=1..9} sin(r*Pi/10)*sin(3*r*Pi/5)*(2*cos(r*Pi/10))^(2*n+1).

a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4).

G.f.: -x^2*(-1+2*x) / ( (x^2-3*x+1)*(5*x^2-5*x+1) ).

a(n+2) = A217593(n,n+5). - Philippe Deléham, Mar 28 2013

2*a(n) = A030191(n-1) - A001906(n). - R. J. Mathar, Nov 15 2019