bibkey: sloane2019a309221 authors: N. J. A. Sloane year: 2019 title: “OEIS A309221, Expected number of distinct squares visited by a knight’s random walk on an infinite chessboard after n steps, normalized to give an integer” doi: null url: https://oeis.org/A309221 claim: “The entry normalizes the expected number of distinct squares visited by a knight’s random walk after n steps (A326954/A326955) as a(n) = E(n)*2^(3n-3) and notes that its integrality is only a conjecture.” strata_touched:

• D5/S3/StatisticalMechanics/RandomWalks/KnightWalkRangeIntegrality license: citation-only triage: anchor

OEIS A309221

The NAME of A309221 is

Expected number of distinct squares visited by a knight’s random walk on an infinite chessboard after n steps, normalized to give an integer.

and its COMMENTS read

Based on A326954/A326955.

a(0)=1; for n>0, a(n) = (A326954(n)/A326955(n))2^(3n-3). (It is only a conjecture that this is always an integer).

A326954 and A326955 (Orson R. L. Peters, 2019) are the numerators and denominators of the expected number of distinct squares visited by a knight’s random walk on an infinite chessboard after n steps, with the comment

The starting square is always considered part of the walk.

They link a Math StackExchange answer (question 3312820, answer 3312917) that computes the expectation with the four-fold rotational symmetry of the board.

Verified locator

• URL: https://oeis.org/A309221
• Version and location: OEIS A309221, revision 7 (2019-08-28), COMMENTS; entry by N. J. A. Sloane, 2019-08-28.
• Related entries: https://oeis.org/A326954, https://oeis.org/A326955.