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

Author

Daniel Poveda Parrilla, Feb 14 2017

Keywords

• nonn

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: + . 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

Examples

			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
		

Links

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

Crossrefs

Cf. A282389.

Programs

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))<>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

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.