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A094811
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Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 1, s(2n+1) = 6.
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4
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1, 6, 26, 100, 364, 1288, 4488, 15504, 53296, 182688, 625184, 2137408, 7303360, 24946816, 85196928, 290926848, 993379072, 3391793664, 11580678656, 39539651584, 134998297600, 460915984384, 1573671536640, 5372862566400, 18344123969536
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OFFSET
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2,2
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COMMENTS
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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+1) counts (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = j, s(2n+1) = k.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 2..1000
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Xavier Gérard Viennot, A Strahler bijection between Dyck paths and planar trees. Formal power series and algebraic combinatorics (Barcelona, 1999). Discrete Math. 246 (2002), no. 1-3, 317--329. MR1887493 (2003b:05013)
Index entries for linear recurrences with constant coefficients, signature (6,-10,4).
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FORMULA
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a(n) = (1/4)*Sum_{r=1..7} sin(r*Pi/8)*sin(r*3*Pi/4)*(2*cos(r*Pi/8))^(2n+1).
G.f.: x^2/((1-2*x)*(1-4*x+2*x^2)).
a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3).
a(n) = A005022(n-2), n>2. - R. J. Mathar, Sep 05 2008
The g.f. x^3/(1 - 6x + 10x^2 - 4x^3) occurs on page 320 of Viennot, 2002.
a(n) = (A006012(n) - 2^n)/2. - R. J. Mathar, Jun 29 2012
a(n) = (-2^(1+n) + (2-sqrt(2))^n + (2+sqrt(2))^n)/4. - Colin Barker, Apr 27 2016
E.g.f.: exp(2*x)*sinh(x/sqrt(2))^2. - Ilya Gutkovskiy, Apr 27 2016
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MATHEMATICA
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CoefficientList[Series[1/((1 - 2x)(1 - 4x + 2x^2)), {x, 0, 200}], x] (* Vincenzo Librandi, Oct 21 2012 *)
Table[FullSimplify[TrigToExp[(1/4) Sum[Sin[r*Pi/8] Sin[3 r Pi/4] (2 Cos[r Pi/8])^(2 n + 1), {r, 7}]]], {n, 2, 26}] (* Michael De Vlieger, Apr 27 2016 *)
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PROG
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(Magma) I:=[1, 6, 26]; [n le 3 select I[n] else 6*Self(n-1) - 10*Self(n-2) + 4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Oct 21 2012
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CROSSREFS
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See A005022 for another version.
Sequence in context: A137746 A344504 A261064 * A005022 A125107 A301476
Adjacent sequences: A094808 A094809 A094810 * A094812 A094813 A094814
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KEYWORD
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nonn,easy
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AUTHOR
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Herbert Kociemba, Jun 11 2004
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EXTENSIONS
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Additional comments from N. J. A. Sloane, May 01 2012
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STATUS
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approved
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