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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
OFFSET
0,3
COMMENTS
The starting square is always considered part of the walk.
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][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
CROSSREFS
See A326954 for numerators. Cf. A309221.
Sequence in context: A268482 A231234 A096687 * A199374 A289538 A154224
KEYWORD
nonn,frac,walk
AUTHOR
Orson R. L. Peters, Aug 08 2019
STATUS
approved