

A282390


Width of polyominoes in A282389.


2



3, 5, 8, 14, 27, 53, 104, 206, 410, 818, 1635, 3269, 6536, 13070, 26139, 52277, 104552, 209102, 418202, 836402, 1672803, 3345605, 6691209, 13382417, 26764832, 53529662, 107059322, 214118642, 428237283, 856474565, 1712949128, 3425898254, 6851796507
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OFFSET

1,1


COMMENTS

Polyominoes in A282389 have got a width of a(n+1) squares and a height of A000051(n) squares.
The polyomino may be represented as a sequence of the lengths of steps in the "ladder" of the polyomino: [2, 1] for Ltetromino, [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


LINKS



FORMULA

a(1) = 3, a(n) = 2*a(n1)  k for n > 1, where k is the width of the central step in the "ladder", which is 1 or 2.


EXAMPLE

a(1) = 3
a(2) = 2 * 3  1 = 5
a(3) = 2 * 5  2 = 8
a(4) = 2 * 8  2 = 14
a(5) = 2 * 14  1 = 27
a(6) = 2 * 27  1 = 53
a(7) = 2 * 53  2 = 104
a(8) = 2 * 104  2 = 206
a(9) = 2 * 206  2 = 410
a(10) = 2 * 410  2 = 818
a(11) = 2 * 818  1 = 1635
a(12) = 2 * 1635  1 = 3269
a(13) = 2 * 3269  2 = 6536
a(14) = 2 * 6536  2 = 13070
a(15) = 2 * 13070  1 = 26139
a(16) = 2 * 26139  1 = 52277
a(17) = 2 * 52277  2 = 104552
a(18) = 2 * 104552  2 = 209102


PROG

(Python)
w, h, bp, bp2 = 3, 2, 0b10, 0b01
for i in range(1, 10):
print(w)
w, h, bp, bp2 = w*2(2 if (bp&1) else 1), 2**i+1, ((bp2&((1<<(h1))1))<<h)+bp2, (bp<<(h1))+(bp>>1)
for i in range(100):
print(w)
w, h, bp, bp2 = w*2(2 if (bp&1) else 1), h1, bp2, (bp>>1)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



