A326954 Numerator of the expected number of distinct squares visited by a knight's random walk on an infinite chessboard after n steps.
1, 2, 23, 15, 2355, 1395, 102971, 58331, 16664147, 9197779, 160882675, 87300443, 48181451689, 25832538281, 881993826001, 468673213505, 508090131646771, 268129446332211, 4514206380211785, 2369170809554097, 317528931045821675
Offset: 0
Examples
a(0) = 1 (from 1/1), we count the starting square. a(1) = 2 (from 2/1), each possible first step is unique. a(2) = 23 (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.
Links
- Math StackExchange, Relatively efficient program to compute a(n) for larger n.
Programs
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Python
from itertools import product from fractions import Fraction def walk(steps): s = [(0, 0)] for dx, dy in steps: s.append((s[-1][0] + dx, s[-1][1] + dy)) return s moves = [(1, 2), (1, -2), (-1, 2), (-1, -2), (2, 1), (2, -1), (-2, 1), (-2, -1)] A326954 = lambda n: Fraction( sum(len(set(walk(steps))) for steps in product(moves, repeat=n)), 8**n ).numerator
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