cp's OEIS Frontend

Each free polyomino represented by a number in A246521 may correspond to 1, 2, 4 or 8 different fixed polyominoes, generated by rotation or reflection.

In the sequence A246521, the size n polyominoes start at position j = 1 + Sum_{i=0..n-1} A000105(i) and end at position k = Sum_{i=0..n} A000105(i). Therefore, the number of fixed polyominoes, A001168(n), is equal to Sum_{i=j..k} a(i).

Given the current set of cells in a diffusion-limited aggregation process on the square lattice, with new cells coming in from infinity, the probability that the next cell appears in a given position can be found by "Spitzer's recipe" (see Spitzer (1976) and Wolf (1991)). These probabilities can then be aggregated to probabilities for each polyomino to appear.

Each row corresponds to a number in the field Q(Pi), i.e., a number of the form (Sum_{i=0..j} p_i*Pi^i)/(Sum_{i=0..k} q_i*Pi^i), with p_i and q_i integers.

Rows A130866(k-1)+1 to A130866(k) correspond to k-celled polyominoes, k >= 2. The sum of the numbers on those rows is 1.

In internal diffusion-limited aggregation on the square lattice, there is one initial cell in the origin. In each subsequent step, a new cell is added by starting a random walk at the origin, adding the first new cell visited. a(n)/A368387(n) is the probability that, when the appropriate number of cells have been added, those cells form the free polyomino with binary code A246521(n+1).

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

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. In the version considered here, all such new squares have the same probability of being selected, whereas in Eden (1961) it appears that the probability is proportional to the number of already existing squares with which the new square shares an edge. See A367760 for the latter version.

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

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.

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

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).

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

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).

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

In internal diffusion-limited aggregation on the square lattice, there is one initial cell in the origin. In each subsequent step, a new cell is added by starting a random walk at the origin, adding the first new cell visited. A368386(n)/a(n) is the probability that, when the appropriate number of cells have been added, those cells form the free polyomino with binary code A246521(n+1).

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

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

Terms on the n-th row are (2*n-1)-smooth.

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