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A344902
Number of open tours by a biased rook on a specific f(n) X 1 board, where f(n) = A070941(n) and cells are colored white or black according to the binary representation of 2n.
3
1, 2, 4, 6, 8, 18, 18, 24, 16, 54, 54, 96, 54, 96, 96, 120, 32, 162, 162, 384, 162, 384, 384, 600, 162, 384, 384, 600, 384, 600, 600, 720, 64, 486, 486, 1536, 486, 1536, 1536, 3000, 486, 1536, 1536, 3000, 1536, 3000, 3000, 4320, 486, 1536, 1536, 3000, 1536
OFFSET
0,2
COMMENTS
A cell is colored white if the binary digit is 0 and a cell is colored black if the binary digit is 1. A biased rook on a white cell moves to the left to any cell or to the right only to a black cell. A biased rook on a black cell moves in any direction.
LINKS
FORMULA
a(n) = A000120(n)!*(1 + A000120(n))^(A023416(n) + 1) for n > 0 with a(0)=1.
a(2n) = (1 + A000120(n))*a(n) for n > 0 with a(0)=1.
From Mikhail Kurkov, Oct 16 2021: (Start)
Conjecture: a(n) = A284005(A073138(n)) for n >= 0 (noticed by Sequence Machine).
Proof: note that A073138(n) in binary is A000120(n) of ones followed by A023416(n) zeros. Then use the formula from "Comments on A284005". (End) [verification needed]
MATHEMATICA
a[n_] := With[{s = DigitCount[n, 2]}, s[[1]]! * (1 + s[[1]])^(1 + s[[2]])]; a[0] = 1; Array[a, 50, 0] (* Amiram Eldar, Aug 03 2023 *)
KEYWORD
nonn,base
AUTHOR
Mikhail Kurkov, Jun 01 2021 [verification needed]
STATUS
approved