A099363 -id:A099363 - cp's OEIS frontend

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

Reinhard Zumkeller, Mar 04 2012

nonn

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

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

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.

Cf. A099363, A208101.

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

a208355 n = a208101 n n
a208355_list = map last a208101_tabl

[Ceiling(Catalan(n div 2)): n in [1..40]]; // Vincenzo Librandi, Feb 18 2014

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

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 *)

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)

A129996 a(n) = (-1)^[(n+1)/2] A000108([n/2]+1).

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Jun 15 2007

sign

Terms of A000108(1,...) repeated, changing sign between the two repeated terms of same magnitude, but keeping the same sign for the next (larger) term. - M. F. Hasler, Aug 25 2012

Cf. A099363, A106181 and A208355, which also consist of duplicated terms of A000108. - M. F. Hasler, Aug 25 2012

A129996(n)=(-1)^((n+1)\2)*A000108(n\2+1) \\ - M. F. Hasler, Aug 25 2012

First differences of A129110: a(n)=A129110(n+1)-A129110(n).

Definition (formula) corrected by M. F. Hasler, Aug 25 2012

A106181 Expansion of c(-x^2)(1+2x-sqrt(1+4x^2))/2, c(x) the g.f. of A000108.

Original entry on oeis.org

0, 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, -1767263190

Offset: 0

Paul Barry, Apr 24 2005

Second column of number triangle A106180.

Cf. A000108, A099363.

a(n) = sin(Pi*n/2)*(C((n-1)/2)*(1-(-1)^n)/2) + sin(Pi*(n+1)/2)*(C(n/2)*(1+(-1)^n)/2) - 0^n for n > 0.

Conjecture: (n+2)*a(n) + n*a(n-1) + 4*(n-1)*a(n-2) + 4*(n-3)*a(n-3) = 0. - R. J. Mathar, Nov 15 2011

cp's OEIS Frontend

A208355 Right edge of the triangle in A208101.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Formula

A129996 a(n) = (-1)^[(n+1)/2] A000108([n/2]+1).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

Extensions

A106181 Expansion of c(-x^2)(1+2x-sqrt(1+4x^2))/2, c(x) the g.f. of A000108.

Views

Author

Keywords

Comments

Crossrefs

Formula

A129110 A transformation of the Catalan sequence.

Views

Author

Keywords

Formula

Extensions