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A367995 a(n) is the denominator of the probability that the free polyomino with binary code A246521(n+1) appears as the image of a simple random walk on the square lattice.

%I A367995 #8 Dec 21 2023 15:37:06
%S A367995 1,1,3,3,21,21,7,21,21,1001,77,77,77,1001,77,77,1001,1001,77,91,77,
%T A367995 89089,785603,143,143,24297,143,25924899,97097,785603,97097,25924899,
%U A367995 143,89089,143,97097,97097,143,25924899,97097,97097,143,143,97097,143,97097,143,143,97097,97097,97097,143,97097,291291,291291,143
%N A367995 a(n) is the denominator of the probability that the free polyomino with binary code A246521(n+1) appears as the image of a simple random walk on the square lattice.
%C A367995 In a simple random walk on the square lattice, draw a unit square around each visited point. A367994(n)/a(n) is the probability that, when the appropriate number of distinct points have been visited, the drawn squares form the free polyomino with binary code A246521(n+1).
%C A367995 Can be read as an irregular triangle, whose n-th row contains A000105(n) terms, n >= 1.
%H A367995 Pontus von Brömssen, <a href="/A367995/b367995.txt">Table of n, a(n) for n = 1..6473</a> (rows 1..10).
%H A367995 <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.
%F A367995 A367994(n)/a(n) = (A368000(n)/A368001(n))*A335573(n+1).
%e A367995 As an irregular triangle:
%e A367995      1;
%e A367995      1;
%e A367995      3,  3;
%e A367995     21, 21,  7, 21,   21;
%e A367995   1001, 77, 77, 77, 1001, 77, 77, 1001, 1001, 77, 91, 77;
%e A367995   ...
%e A367995 There are only one monomino and one free domino, so both of these appear with probability 1, and a(1) = a(2) = 1.
%e A367995 For three squares, the probability for an L (or right) tromino (whose binary code is 7 = A246521(4)) is 2/3, so a(3) = 3. The probability for the straight tromino (whose binary code is 11 = A246521(5)) is 1/3, so a(4) = 3.
%Y A367995 Cf. A000105, A246521, A335573, A367672, A367761, A367994 (numerators), A367997, A367999, A368000, A368001.
%K A367995 nonn,frac,tabf
%O A367995 1,3
%A A367995 _Pontus von Brömssen_, Dec 08 2023