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A094811 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.

%I A094811 #39 Feb 11 2025 10:31:46
%S A094811 1,6,26,100,364,1288,4488,15504,53296,182688,625184,2137408,7303360,
%T A094811 24946816,85196928,290926848,993379072,3391793664,11580678656,
%U A094811 39539651584,134998297600,460915984384,1573671536640,5372862566400,18344123969536
%N A094811 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.
%C A094811 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.
%H A094811 Vincenzo Librandi, <a href="/A094811/b094811.txt">Table of n, a(n) for n = 2..1000</a>
%H A094811 Andrei Asinowski and Michaela A. Polley, <a href="https://arxiv.org/abs/2501.11781">Patterns in rectangulations. Part I: T-like patterns, inversion sequence classes I(010, 101, 120, 201) and I(011, 201), and rushed Dyck paths</a>, arXiv:2501.11781 [math.CO], 2025. See p. 26.
%H A094811 G. Kreweras, <a href="/A000108/a000108_1.pdf">Sur les éventails de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]
%H A094811 László Németh and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Nemeth/nemeth8.html">Sequences Involving Square Zig-Zag Shapes</a>, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
%H A094811 Xavier Gérard Viennot, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00265-5">A Strahler bijection between Dyck paths and planar trees</a>. Formal power series and algebraic combinatorics (Barcelona, 1999). Discrete Math. 246 (2002), no. 1-3, 317--329. MR1887493 (2003b:05013)
%H A094811 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-10,4).
%F A094811 a(n) = (1/4)*Sum_{r=1..7} sin(r*Pi/8)*sin(r*3*Pi/4)*(2*cos(r*Pi/8))^(2n+1).
%F A094811 G.f.: x^2/((1-2*x)*(1-4*x+2*x^2)).
%F A094811 a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3).
%F A094811 a(n) = A005022(n-2), n>2. - _R. J. Mathar_, Sep 05 2008
%F A094811 The g.f. x^3/(1 - 6x + 10x^2 - 4x^3) occurs on page 320 of Viennot, 2002.
%F A094811 a(n) = (A006012(n) - 2^n)/2. - _R. J. Mathar_, Jun 29 2012
%F A094811 a(n) = (-2^(1+n) + (2-sqrt(2))^n + (2+sqrt(2))^n)/4. - _Colin Barker_, Apr 27 2016
%F A094811 E.g.f.: exp(2*x)*sinh(x/sqrt(2))^2. - _Ilya Gutkovskiy_, Apr 27 2016
%t A094811 CoefficientList[Series[1/((1 - 2x)(1 - 4x + 2x^2)), {x, 0, 200}], x] (* _Vincenzo Librandi_, Oct 21 2012 *)
%t A094811 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 *)
%o A094811 (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
%Y A094811 See A005022 for another version.
%K A094811 nonn,easy
%O A094811 2,2
%A A094811 _Herbert Kociemba_, Jun 11 2004
%E A094811 Additional comments from _N. J. A. Sloane_, May 01 2012