A282390 Width of polyominoes in A282389.

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

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 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

Links

Andrey Zabolotskiy, Table of n, a(n) for n = 1..3300

Formula

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.

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<<(h-1))-1))<<h)+bp2, (bp<<(h-1))+(bp>>1)
for i in range(100):
    print(w)
    w, h, bp, bp2 = w*2-(2 if (bp&1) else 1), h-1, bp2, (bp>>1)
# Andrey Zabolotskiy, Feb 14 2017

Crossrefs

Cf. A282389.

Sequence in context: A175378 A072655 A108301 * A095290 A080999 A077579

Adjacent sequences: A282387 A282388 A282389 * A282391 A282392 A282393

Keyword

nonn

Author

Daniel Poveda Parrilla, Feb 14 2017

Status

approved