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A282390 Width of polyominoes in A282389.

%I A282390 #21 Feb 14 2017 22:35:22
%S A282390 3,5,8,14,27,53,104,206,410,818,1635,3269,6536,13070,26139,52277,
%T A282390 104552,209102,418202,836402,1672803,3345605,6691209,13382417,
%U A282390 26764832,53529662,107059322,214118642,428237283,856474565,1712949128,3425898254,6851796507
%N A282390 Width of polyominoes in A282389.
%C A282390 Polyominoes in A282389 have got a width of a(n+1) squares and a height of A000051(n) squares.
%C A282390 The polyomino may be represented as a sequence of the lengths of steps in the "ladder" of the polyomino: [2, 1] for L-tetromino, [2, 1, 2] for the next iteration, and so on. The overall width is the sum of these lengths. And on the next iteration, the new sequence of lengths of steps is formed from the previous one as: <previous sequence, reversed, with the first (after reversion) element removed> + <previous sequence, reversed>. So the sequence always consists of 1's and 2's only and therefore can be encoded as a binary string of length 2^n+1. This is exploited in the Python program below and explains the formula. - _Andrey Zabolotskiy_, Feb 14 2017
%H A282390 Andrey Zabolotskiy, <a href="/A282390/b282390.txt">Table of n, a(n) for n = 1..3300</a>
%F A282390 a(1) = 3, a(n) = 2*a(n-1) - k for n > 1, where k is the width of the central step in the "ladder", which is 1 or 2.
%e A282390 a(1) = 3
%e A282390 a(2) = 2 * 3 - 1 = 5
%e A282390 a(3) = 2 * 5 - 2 = 8
%e A282390 a(4) = 2 * 8 - 2 = 14
%e A282390 a(5) = 2 * 14 - 1 = 27
%e A282390 a(6) = 2 * 27 - 1 = 53
%e A282390 a(7) = 2 * 53 - 2 = 104
%e A282390 a(8) = 2 * 104 - 2 = 206
%e A282390 a(9) = 2 * 206 - 2 = 410
%e A282390 a(10) = 2 * 410 - 2 = 818
%e A282390 a(11) = 2 * 818 - 1 = 1635
%e A282390 a(12) = 2 * 1635 - 1 = 3269
%e A282390 a(13) = 2 * 3269 - 2 = 6536
%e A282390 a(14) = 2 * 6536 - 2 = 13070
%e A282390 a(15) = 2 * 13070 - 1 = 26139
%e A282390 a(16) = 2 * 26139 - 1 = 52277
%e A282390 a(17) = 2 * 52277 - 2 = 104552
%e A282390 a(18) = 2 * 104552 - 2 = 209102
%o A282390 (Python)
%o A282390 w, h, bp, bp2 = 3, 2, 0b10, 0b01
%o A282390 for i in range(1, 10):
%o A282390     print(w)
%o A282390     w, h, bp, bp2 = w*2-(2 if (bp&1) else 1), 2**i+1, ((bp2&((1<<(h-1))-1))<<h)+bp2, (bp<<(h-1))+(bp>>1)
%o A282390 for i in range(100):
%o A282390     print(w)
%o A282390     w, h, bp, bp2 = w*2-(2 if (bp&1) else 1), h-1, bp2, (bp>>1)
%o A282390 # _Andrey Zabolotskiy_, Feb 14 2017
%Y A282390 Cf. A282389.
%K A282390 nonn
%O A282390 1,1
%A A282390 _Daniel Poveda Parrilla_, Feb 14 2017