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 Christersson 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, Séminaire Lotharingien de Combinatoire 84B (2020), Article #95; see also
Malin Christersson, Make hyperbolic tilings of images, web page, 2019.
Derek Levin, Lara Pudwell, Manda Riehl, and Andrew Sandberg, Pattern Avoidance on k-ary Heaps, Slides of Talk, 2014. [Wayback Machine link]
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) = 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.
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)
a(n) ~ c * 2^(n+3/2) / (n^(3/2) * sqrt(Pi)), where c = 2 if n is odd and c = 1 if n is even. - Amiram Eldar, Sep 16 2025
a(n) = A008315(n,floor(n/2)). - Alois P. Heinz, Mar 11 2026
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
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Mar 04 2012
STATUS
approved