A094831 - OEIS

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A094831

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) = 3.

1, 2, 6, 19, 62, 207, 703, 2417, 8382, 29242, 102431, 359790, 1266103, 4460939, 15730497, 55500634, 195890270, 691566411, 2441886670, 8623112591, 30453261927, 107553444913, 379864424726, 1341658806066, 4738726458775

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OFFSET

0,2

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.

A comparison of their recurrence relations shows that this sequence is the even bisection of A188048. - John Blythe Dobson, Jun 20 2015

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..1825

Index entries for linear recurrences with constant coefficients, signature (6,-9,1)

FORMULA

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

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

G.f.: (1-4*x+3*x^2)/(1-6*x+9*x^2-x^3).

MATHEMATICA

CoefficientList[Series[(1 - 4 x + 3 x^2)/(1 - 6 x + 9 x^2 - x^3), {x, 0, 24}], x] (* Michael De Vlieger, Feb 12 2022 *)

PROG

(PARI) Vec((1-4*x+3*x^2)/(1-6*x+9*x^2-x^3) + O(x^30)) \\ Michel Marcus, Jun 21 2015