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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.

%I A344902 #33 Apr 21 2024 22:27:34
%S A344902 1,2,4,6,8,18,18,24,16,54,54,96,54,96,96,120,32,162,162,384,162,384,
%T A344902 384,600,162,384,384,600,384,600,600,720,64,486,486,1536,486,1536,
%U A344902 1536,3000,486,1536,1536,3000,1536,3000,3000,4320,486,1536,1536,3000,1536
%N 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.
%C A344902 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.
%H A344902 Amiram Eldar, <a href="/A344902/b344902.txt">Table of n, a(n) for n = 0..10000</a>
%H A344902 Jon Maiga, <a href="http://sequencedb.net/s/A344902">Computer-generated formulas for A344902</a>, Sequence Machine.
%F A344902 a(n) = A000120(n)!*(1 + A000120(n))^(A023416(n) + 1) for n > 0 with a(0)=1.
%F A344902 a(2n) = (1 + A000120(n))*a(n) for n > 0 with a(0)=1.
%F A344902 From _Mikhail Kurkov_, Oct 16 2021: (Start)
%F A344902 Conjecture: a(n) = A284005(A073138(n)) for n >= 0 (noticed by Sequence Machine).
%F A344902 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]
%t A344902 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 *)
%Y A344902 Cf. A000120, A023416, A070941, A073138, A284005, A329369, A329718.
%K A344902 nonn,base
%O A344902 0,2
%A A344902 _Mikhail Kurkov_, Jun 01 2021 [verification needed]