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

0,2

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

Amiram Eldar, Table of n, a(n) for n = 0..10000

Jon Maiga, Computer-generated formulas for A344902, Sequence Machine.

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 *)

Crossrefs

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

Sequence in context: A100055 A334031 A304660 * A104001 A048784 A152491

Adjacent sequences: A344899 A344900 A344901 * A344903 A344904 A344905

Keyword

nonn,base

Author

Mikhail Kurkov, Jun 01 2021 [verification needed]

Status

approved