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A329718 The 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 A329718 #37 Apr 21 2024 22:26:51
%S A329718 1,2,4,4,8,6,14,8,16,10,24,10,46,24,46,16,32,18,44,14,84,34,68,18,146,
%T A329718 68,138,44,230,84,146,32,64,34,84,22,160,54,112,22,276,106,224,54,376,
%U A329718 106,192,34,454,192,406,112,690,224,406,84,1066,376,690,160
%N A329718 The 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 A329718 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 only to the left and otherwise moves only to the right.
%H A329718 Mikhail Kurkov, <a href="/A329718/a329718_1.txt">Comments on A329718</a> [verification needed]
%F A329718 a(n) = f(n) + f(A059894(n)) = f(n) + f(2*A053645(n)) for n > 0 with a(0) = 1 where f(n) = A329369(n).
%F A329718 Sum_{k=0..2^n-1} a(k) = 2*(n+1)! - 1 for n >= 0.
%F A329718 a((4^n-1)/3) = 2*A110501(n+1) for n > 0.
%F A329718 a(2^1*(2^n-1)) = A027649(n),
%F A329718 a(2^2*(2^n-1)) = A027650(n),
%F A329718 a(2^3*(2^n-1)) = A027651(n),
%F A329718 a(2^4*(2^n-1)) = A283811(n),
%F A329718 and more generally, a(2^m*(2^n-1)) = T(n,m+1) for n >= 0, m >= 0 where T(n,m) = Sum_{k=0..n} k!*(k+1)^m*Stirling2(n,k)*(-1)^(n-k).
%e A329718 a(1) = 2 because the binary expansion of 2 is 10 and there are 2 open biased rook's tours, namely 12 and 21.
%e A329718 a(2) = 4 because the binary expansion of 4 is 100 and there are 4 open biased rook's tours, namely 132, 213, 231 and 321.
%e A329718 a(3) = 4 because the binary expansion of 6 is 110 and there are 4 open biased rook's tours, namely 123, 132, 231 and 312.
%Y A329718 Cf. A027649, A027650, A027651, A059894, A110501, A283811, A284005, A329369, A344902, A344947.
%K A329718 nonn
%O A329718 0,2
%A A329718 _Mikhail Kurkov_, Nov 19 2019 [verification needed]