A326955 Denominator of the expected number of distinct squares visited by a knight's random walk on an infinite chessboard after n steps.
1, 1, 8, 4, 512, 256, 16384, 8192, 2097152, 1048576, 16777216, 8388608, 4294967296, 2147483648, 68719476736, 34359738368, 35184372088832, 17592186044416, 281474976710656, 140737488355328, 18014398509481984
Offset: 0
Examples
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.
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)] A326955 = lambda n: Fraction( sum(len(set(walk(steps))) for steps in product(moves, repeat=n)), 8**n ).denominator
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