A208355 Right edge of the triangle in A208101.

1, 1, 1, 2, 2, 5, 5, 14, 14, 42, 42, 132, 132, 429, 429, 1430, 1430, 4862, 4862, 16796, 16796, 58786, 58786, 208012, 208012, 742900, 742900, 2674440, 2674440, 9694845, 9694845, 35357670, 35357670, 129644790, 129644790, 477638700, 477638700, 1767263190

Offset

0,4

Comments

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

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).

Malin Christensson, Make hyperbolic tilings of images, web page, 2019.

D. Levin, L. Pudwell, M. Riehl, and A. Sandberg, Pattern Avoidance on k-ary Heaps, Slides of Talk, 2014.

Zhicong Lin, David G. L. Wang, and Tongyuan Zhao, A decomposition of ballot permutations, pattern avoidance and Gessel walks, arXiv:2103.04599 [math.CO], 2021.

Formula

a(n) = A000108(floor((n+1)/2)), where A000108 = Catalan numbers.

a(n) = A208101(n,n).

a(n) = abs(A099363(n)).

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

From Robert A. Russell, Jan 19 2024: (Start)

a(2m) = C(2m,m)/(m+1); a(2m-1) = a(2m); a(n+2)/a(n) ~ 4.

a(n-1) = 2*A000207(n) - A001683(n+2) = A001683(n+2) - 2*A369314(n) = A000207(n) - A369314(n). (End)

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

G.f.: ((((1+z)*(1-sqrt(1-4*z^2)))/(2*z^2))-1)/z. - Robert A. Russell, Jan 28 2024

From Peter Bala, Feb 05 2024: (Start)

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.

a(n) = Sum_{k = 0..n} (-2)^(n-k)*binomial(n, k)*A000245(k+1).

a(n) = (-2)^n * hypergeom([-n, 3/2, 2], [1, 4], 2). (End)

Example

a(0)=1; a(1)=1; a(2)=1; a(3)=2. - Robert A. Russell, Jan 19 2024
____      ________
\  /  /\  \  /\  /  /\     /\
 \/  /__\  \/__\/  /__\   /__\____
     \  /         /\  /\  \  /\  /
      \/         /__\/__\  \/__\/

Maple

A208355_list := proc(len) local D, b, h, R, i, k;
    D := [seq(0, j=0..len+2)]; D[1] := 1; b := true; h := 2; R := NULL;
    for i from 1 to 2*len do
        if b then
            for k from h by -1 to 2 do D[k] := D[k] - D[k-1] od;
            h := h + 1; R := R, abs(D[2]);
        else
            for k from 1 by 1 to h do D[k] := D[k] + D[k+1] od;
        fi;
        b := not b:
    od;
    return R
end:
A208355_list(38); # Peter Luschny, Dec 19 2017

Mathematica

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];
a[n_] := T[n, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 03 2018, from A208101 *)
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 *)

Prog

(Haskell)
a208355 n = a208101 n n
a208355_list = map last a208101_tabl
(Magma) [Ceiling(Catalan(n div 2)): n in [1..40]]; // Vincenzo Librandi, Feb 18 2014

Crossrefs

Cf. A099363, A208101.

Polyominoes: A001683(n+2) (oriented), A000207 (unoriented). A369314 (chiral), A000108 (rooted), A047749 ({4,oo}.

Sequence in context: A285013 A095014 A129996 * A099363 A106181 A098887

Adjacent sequences: A208352 A208353 A208354 * A208356 A208357 A208358

Keyword

nonn

Author

Reinhard Zumkeller, Mar 04 2012

Status

approved