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A208355 Right edge of the triangle in A208101.

%I A208355 #67 Feb 18 2025 03:51:26
%S A208355 1,1,1,2,2,5,5,14,14,42,42,132,132,429,429,1430,1430,4862,4862,16796,
%T A208355 16796,58786,58786,208012,208012,742900,742900,2674440,2674440,
%U A208355 9694845,9694845,35357670,35357670,129644790,129644790,477638700,477638700,1767263190
%N A208355 Right edge of the triangle in A208101.
%C A208355 Number of achiral polyominoes composed of n+1 triangular cells of the hyperbolic regular tiling with Schläfli symbol {3,oo}. A stereographic projection of this tiling on the Poincaré disk can be obtained via the Christensson link. An achiral polyomino is identical to its reflection. - _Robert A. Russell_, Jan 20 2024
%H A208355 Reinhard Zumkeller, <a href="/A208355/b208355.txt">Table of n, a(n) for n = 0..1000</a>
%H A208355 Andrei Asinowski, Cyril Banderier, and Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, (2019).
%H A208355 Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.
%H A208355 D. Levin, L. Pudwell, M. Riehl, and A. Sandberg, <a href="http://www.etsu.edu/cas/math/pp2014/documents/talks/riehl.pdf">Pattern Avoidance on k-ary Heaps</a>, Slides of Talk, 2014.
%H A208355 Zhicong Lin, David G. L. Wang, and Tongyuan Zhao, <a href="https://arxiv.org/abs/2103.04599">A decomposition of ballot permutations, pattern avoidance and Gessel walks</a>, arXiv:2103.04599 [math.CO], 2021.
%F A208355 a(n) = A000108(floor((n+1)/2)), where A000108 = Catalan numbers.
%F A208355 a(n) = A208101(n,n).
%F A208355 a(n) = abs(A099363(n)).
%F A208355 Conjecture: -(n+3)*(n-2)*a(n) - 4*a(n-1) + 4*(n-1)^2*a(n-2) = 0. - _R. J. Mathar_, Aug 04 2015
%F A208355 From _Robert A. Russell_, Jan 19 2024: (Start)
%F A208355 a(2m) = C(2m,m)/(m+1); a(2m-1) = a(2m); a(n+2)/a(n) ~ 4.
%F A208355 a(n-1) = 2*A000207(n) - A001683(n+2) = A001683(n+2) - 2*A369314(n) = A000207(n) - A369314(n). (End)
%F A208355 G.f.: (G(z^2)+z*G(z^2)-1)/z, where G(z)=1+z*G(z)^2, the generating function for A000108. - _Robert A. Russell_, Jan 26 2024
%F A208355 G.f.: ((((1+z)*(1-sqrt(1-4*z^2)))/(2*z^2))-1)/z. - _Robert A. Russell_, Jan 28 2024
%F A208355 From _Peter Bala_, Feb 05 2024: (Start)
%F A208355 G.f.: 1/(1 + 2*x) * c(x/(1 + 2*x))^3, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
%F A208355 a(n) = Sum_{k = 0..n} (-2)^(n-k)*binomial(n, k)*A000245(k+1).
%F A208355 a(n) = (-2)^n * hypergeom([-n, 3/2, 2], [1, 4], 2). (End)
%e A208355 a(0)=1; a(1)=1; a(2)=1; a(3)=2. - _Robert A. Russell_, Jan 19 2024
%e A208355 ____      ________
%e A208355 \  /  /\  \  /\  /  /\     /\
%e A208355  \/  /__\  \/__\/  /__\   /__\____
%e A208355      \  /         /\  /\  \  /\  /
%e A208355       \/         /__\/__\  \/__\/
%p A208355 A208355_list := proc(len) local D, b, h, R, i, k;
%p A208355     D := [seq(0, j=0..len+2)]; D[1] := 1; b := true; h := 2; R := NULL;
%p A208355     for i from 1 to 2*len do
%p A208355         if b then
%p A208355             for k from h by -1 to 2 do D[k] := D[k] - D[k-1] od;
%p A208355             h := h + 1; R := R, abs(D[2]);
%p A208355         else
%p A208355             for k from 1 by 1 to h do D[k] := D[k] + D[k+1] od;
%p A208355         fi;
%p A208355         b := not b:
%p A208355     od;
%p A208355     return R
%p A208355 end:
%p A208355 A208355_list(38); # _Peter Luschny_, Dec 19 2017
%t A208355 T[_, 0] = 1; T[n_, 1] := n; T[n_, n_] := T[n - 1, n - 2]; T[n_, k_] /; 1 < k < n := T[n, k] = T[n - 1, k] + T[n - 1, k - 2];
%t A208355 a[n_] := T[n, n];
%t A208355 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Feb 03 2018, from A208101 *)
%t A208355 Table[If[EvenQ[n], Binomial[n,n/2]/(n/2+1), Binomial[n+1,(n+1)/2]/((n+3)/2)], {n,0,40}] (* _Robert A. Russell_, Jan 19 2024 *)
%o A208355 (Haskell)
%o A208355 a208355 n = a208101 n n
%o A208355 a208355_list = map last a208101_tabl
%o A208355 (Magma) [Ceiling(Catalan(n div 2)): n in [1..40]]; // _Vincenzo Librandi_, Feb 18 2014
%Y A208355 Cf. A099363, A208101.
%Y A208355 Polyominoes: A001683(n+2) (oriented), A000207 (unoriented). A369314 (chiral), A000108 (rooted), A047749 ({4,oo}).
%K A208355 nonn
%O A208355 0,4
%A A208355 _Reinhard Zumkeller_, Mar 04 2012