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A094826 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) = 1, s(2n) = 3. 4
1, 3, 9, 28, 90, 297, 1000, 3417, 11799, 41041, 143472, 503262, 1769365, 6230304, 21960801, 77461435, 273351705, 964918116, 3406804786, 12029917377, 42483179304, 150036624217, 529901048943, 1871559855009, 6610286313784 (list; graph; refs; listen; history; text; internal format)
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)(2Cos(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..1826

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 (6,-9,1).

FORMULA

a(n)=(2/9)*Sum(r, 1, 8, Sin(r*Pi/9)Sin(3*r*Pi/9)(2Cos(r*Pi/9))^(2n))

a(n)=6a(n-1)-9a(n-2)+a(n-3) =7a(n-1)-15a(n-2)+10a(n-3)-a(n-4).

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

a(n) = A094829(n+1) -3*A094829(n). - R. J. Mathar, Nov 14 2019

MATHEMATICA

Rest@ CoefficientList[Series[x (-1 + 3 x)/(-1 + 6 x - 9 x^2 + x^3), {x, 0, 25}], x] (* Michael De Vlieger, Aug 05 2021 *)

LinearRecurrence[{6, -9, 1}, {1, 3, 9}, 30] (* Harvey P. Dale, Dec 29 2021 *)

CROSSREFS

Sequence in context: A007822 A094164 A094803 * A033190 A071724 A000245

Adjacent sequences:  A094823 A094824 A094825 * A094827 A094828 A094829

KEYWORD

nonn,easy

AUTHOR

Herbert Kociemba, Jun 13 2004

STATUS

approved

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Last modified January 18 09:26 EST 2022. Contains 350454 sequences. (Running on oeis4.)