A073913 - OEIS

%I #20 Dec 27 2024 14:55:48

%S 1,12,94,604,3461,18412,93016,452500,2139230,9890404,44921002,

%T 201099320,889594210,3896177956,16920602244,72954802376,312595497011,

%U 1332153819572,5650155211024,23864065957572,100418115489408

%N Number of staircase polygons on the square lattice with perimeter 2n and one (possibly rotated) staircase polygonal hole.

%C The old entry with this A-number was a duplicate of A070844.

%H Iwan Jensen, <a href="/A073913/b073913.txt">Table of n, a(n) for n = 8..125</a> [Data from web page]

%H Iwan Jensen, <a href="http://www.ms.unimelb.edu.au/~iwan/polygons/Polygons_ser.html">Polygon enumerations.</a>

%H Iwan Jensen and Andrew Rechnitzer, <a href="http://dx.doi.org/10.1088/1751-8113/41/21/215002">The exact perimeter generating function for a model of punctured staircase polygons</a>, J. Phys. A: Math. Theor. 41 (2008) 215002, Table 1.

%F G.f.: -(1/4)*(f1(x)-f2(x)+f3(x)-f4(x)) where f1(x) = (1-8*x+16*x^2-4*x^3)/(1-4*x), f2(x) = (1-6*x+6*x^2)/sqrt(1-4*x), f3(x) = (1/sqrt(2))*(sqrt(2+sqrt(3+4*x))*(3-8*x+2*x^2-sqrt(3+4*x)*(1-2*x)))/(1-4*x)^(3/4), f4(x) = (1/sqrt(2))*((3-8*x+2*x^2+sqrt(3+4*x)*(1-2*x)))/(1-4*x)^(1/4)/sqrt(2+sqrt(3+4*x)) [from Jensen and Rechnitzer, 2008]. - _Sean A. Irvine_, Dec 27 2024

%Y Cf. A057410, A057414.

%K nonn

%O 8,2

%A _Olivier Gérard_, Feb 14 2009, based on data from the web site of Iwan Jensen.

%E Offset corrected by _Sean A. Irvine_, Dec 27 2024