A367760 - OEIS

%I #6 Dec 03 2023 09:12:18

%S 1,1,2,1,1,1,1,1,1,2,1,1,1,7,1,1,7,7,1,1,1,23,49,1,1,53,1,107,1,49,1,

%T 107,1,23,1,1,1,1,137,1,1,1,1,1,1,1,1,1,1,1,1,1,1,11,7,1,2797,70037,

%U 70037,31,31,2797,3517,1,41,653,49541,1,3517,71,67,41,899,2797,653,1,1,1,1,653,1,1

%N a(n) is the numerator of the probability that the free polyomino with binary code A246521(n+1) appears in the Eden growth model on the square lattice, when n square cells have been added.

%C In the Eden growth model, there is a single initial unit square cell in the plane and more squares are added one at a time, selected randomly among those squares that share an edge with one of the already existing squares, with probabilities proportional to the number of already existing squares with which the new square shares an edge. This seems to be the version described in Eden (1961). See A367671 for another version.

%C Can be read as an irregular triangle, whose n-th row contains A000105(n) terms, n >= 1.

%H Murray Eden, <a href="https://projecteuclid.org/ebooks/berkeley-symposium-on-mathematical-statistics-and-probability/Proceedings-of-the-Fourth-Berkeley-Symposium-on-Mathematical-Statistics-and/chapter/A-Two-dimensional-Growth-Process/bsmsp/1200512888">A two-dimensional growth process</a>, in: 4th Berkeley Symposium on Mathematical Statistics and Probability (Berkeley 1960), vol. 4, pp. 223-239, University of California Press, Berkeley, 1961.

%H <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.

%F a(n)/A367761(n) = (A367764(n)/A367765(n))*A335573(n+1).

%e As an irregular triangle:

%e   1;

%e   1;

%e   2, 1;

%e   1, 1, 1, 1, 1;

%e   2, 1, 1, 1, 7, 1, 1, 7, 7, 1, 1, 1;

%e   ...

%e For n = 7, the T-tetromino has binary code A246521(n+1) = 27. It can be obtained either via the straight tromino (probability 1/3 * 1/4) or via the L-tromino (probability 2/3 * 1/4), so the probability of obtaining the T-tetromino is 1/12 + 1/6 = 1/4 and a(7) = 1.

%Y Cf. A000105, A246521, A335573, A367671, A367761 (denominators), A367762, A367764, A367765.

%K nonn,frac,tabf

%O 1,3

%A _Pontus von Brömssen_, Dec 02 2023