A094821 - OEIS

A094821

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) = 5.

1, 4, 15, 54, 190, 660, 2276, 7816, 26776, 91600, 313104, 1069728, 3653728, 12477504, 42606656, 145479808, 496722304, 1695962368, 5790470400, 19770087936, 67499673088, 230459040768, 786837865472, 2686435477504

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OFFSET

1,2

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).

FORMULA

a(n) = (1/4)*Sum_{r=1..7} sin(3*r*Pi/8)*sin(5*r*Pi/8)*(2*cos(r*Pi/8))^(2n).

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

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

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

E.g.f.: (exp(2*x)*(2*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x) - 1) - 1)/4. - Stefano Spezia, Apr 25 2023

MATHEMATICA

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

CROSSREFS