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.

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

Author

Mikhail Kurkov, Jun 01 2021 [verification needed]

Keywords

• nonn
• base

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

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• Jon Maiga, Computer-generated formulas for A344902, Sequence Machine.

Crossrefs

Cf. A000120, A023416, A070941, A073138, A284005, A329369, A329718.

Programs

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 *)

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]