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A094828 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) = 5. 3
1, 5, 20, 75, 274, 988, 3536, 12597, 44745, 158632, 561683, 1987154, 7026408, 24835744, 87763945, 310088381, 1095490524, 3869911659, 13670143618, 48287147300, 170561502896, 602454835293, 2127962632993, 7516243783216 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

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))^(2*n)) 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 = 2..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 (7,-15,10,-1).

FORMULA

a(n) = (2/9)*sum(r=1..8, sin(r*Pi/9)*sin(5*r*Pi/9)*(2*cos(r*Pi/9))^(2*n)).

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

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

a(n) = A094256(n-1) - 2*A094256(n-2). - R. J. Mathar, Nov 14 2019

3*a(n) = A094829(n+2) -5*A094829(n+1)+7*A094829(n)-1. - R. J. Mathar, Nov 14 2019

MATHEMATICA

LinearRecurrence[{7, -15, 10, -1}, {1, 5, 20, 75}, 30] (* Harvey P. Dale, Apr 27 2020 *)

CROSSREFS

Sequence in context: A248326 A022633 A092490 * A030191 A093131 A224422

Adjacent sequences:  A094825 A094826 A094827 * A094829 A094830 A094831

KEYWORD

nonn,easy

AUTHOR

Herbert Kociemba, Jun 13 2004

STATUS

approved

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Last modified January 27 13:28 EST 2022. Contains 350607 sequences. (Running on oeis4.)