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 A326955 Denominator of the expected number of distinct squares visited by a knight's random walk on an infinite chessboard after n steps. 3
 1, 1, 8, 4, 512, 256, 16384, 8192, 2097152, 1048576, 16777216, 8388608, 4294967296, 2147483648, 68719476736, 34359738368, 35184372088832, 17592186044416, 281474976710656, 140737488355328, 18014398509481984 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The starting square is always considered part of the walk. LINKS Math StackExchange, Relatively efficient program to compute a(n) for larger n. EXAMPLE a(0) = 1 (from 1/1), we count the starting square. a(1) = 1 (from 2/1), each possible first step is unique. a(2) = 8 (from 23/8), as for each possible first step 1/8th of the second steps go back to a previous square, thus the expected distinct squares visited is 2 + 7/8 = 23/8. PROG (Python) from itertools import product from fractions import Fraction def walk(steps):     s = [(0, 0)]     for dx, dy in steps:         s.append((s[-1] + dx, s[-1] + dy))     return s moves = [(1, 2), (1, -2), (-1, 2), (-1, -2),          (2, 1), (2, -1), (-2, 1), (-2, -1)] A326955 = lambda n: Fraction(         sum(len(set(walk(steps)))             for steps in product(moves, repeat=n)),         8**n     ).denominator CROSSREFS See A326954 for numerators. Cf. A309221. Sequence in context: A268482 A231234 A096687 * A199374 A289538 A154224 Adjacent sequences:  A326952 A326953 A326954 * A326956 A326957 A326958 KEYWORD nonn,frac,walk AUTHOR Orson R. L. Peters, Aug 08 2019 STATUS approved

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Last modified May 21 03:31 EDT 2022. Contains 353887 sequences. (Running on oeis4.)