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A367994 a(n) is the numerator 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 A367994 #15 Dec 23 2023 09:46:55
%S A367994 1,1,2,1,8,4,1,4,2,388,4,4,8,64,8,4,32,64,4,1,2,3468,76520,4,4,2495,4,
%T A367994 2102248,1556,76520,1556,1051124,4,3468,4,1194,1556,4,1262762,597,
%U A367994 1556,2,4,1556,4,597,2,2,778,1194,1556,2,1194,2501,1648,1,5270,13652575732976,13652575732976,4468,4468
%N A367994 a(n) is the numerator 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 A367994 In a simple random walk on the square lattice, draw a unit square around each visited point. a(n)/A367995(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 A367994 Can be read as an irregular triangle, whose n-th row contains A000105(n) terms, n >= 1.
%H A367994 Pontus von Brömssen, <a href="/A367994/b367994.txt">Table of n, a(n) for n = 1..6473</a> (rows 1..10).
%H A367994 <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.
%F A367994 a(n)/A367995(n) = (A368000(n)/A368001(n))*A335573(n+1).
%e A367994 As an irregular triangle:
%e A367994     1;
%e A367994     1;
%e A367994     2, 1;
%e A367994     8, 4, 1, 4,  2;
%e A367994   388, 4, 4, 8, 64, 8, 4, 32, 64, 4, 1, 2;
%e A367994   ...
%e A367994 There are only one monomino and one free domino, so both of these appear with probability 1, and a(1) = a(2) = 1.
%e A367994 For three squares, the probability for an L (or right) tromino (whose binary code is 7 = A246521(4)) is 2/3, so a(3) = 2. The probability for the straight tromino (whose binary code is 11 = A246521(5)) is 1/3, so a(4) = 1.
%Y A367994 Cf. A000105, A246521, A335573, A367671, A367760, A367995 (denominators), A367996, A367998, A368000, A368001, A368386.
%K A367994 nonn,frac,tabf
%O A367994 1,3
%A A367994 _Pontus von Brömssen_, Dec 08 2023