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.

1, 1, 3, 3, 21, 21, 7, 21, 21, 1001, 77, 77, 77, 1001, 77, 77, 1001, 1001, 77, 91, 77, 89089, 785603, 143, 143, 24297, 143, 25924899, 97097, 785603, 97097, 25924899, 143, 89089, 143, 97097, 97097, 143, 25924899, 97097, 97097, 143, 143, 97097, 143, 97097, 143, 143, 97097, 97097, 97097, 143, 97097, 291291, 291291, 143

Offset

1,3

Comments

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.

Links

Pontus von Brömssen, Table of n, a(n) for n = 1..6473 (rows 1..10).

Index entries for sequences related to polyominoes.

Formula

A367994(n)/a(n) = (A368000(n)/A368001(n))*A335573(n+1).

Example

As an irregular triangle:
     1;
     1;
     3,  3;
    21, 21,  7, 21,   21;
  1001, 77, 77, 77, 1001, 77, 77, 1001, 1001, 77, 91, 77;
  ...
There are only one monomino and one free domino, so both of these appear with probability 1, and a(1) = a(2) = 1.
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.

Crossrefs

Cf. A000105, A246521, A335573, A367672, A367761, A367994 (numerators), A367997, A367999, A368000, A368001.

Sequence in context: A292221 A112534 A006656 * A205452 A159910 A172485

Adjacent sequences: A367992 A367993 A367994 * A367996 A367997 A367998

Keyword

nonn,frac,tabf

Author

Pontus von Brömssen, Dec 08 2023

Status

approved