A165197 Smallest integers whose binary digits "1" define a polyomino if sorted into a quadrant shape.

1, 3, 7, 11, 15, 23, 27, 30, 31, 47, 62, 63, 75, 79, 91, 94, 95, 111, 126, 127, 143, 159, 175, 181, 182, 183, 188, 189, 190, 191, 207, 219, 220, 221, 222, 223, 239, 252, 253, 254, 255, 347, 350, 351, 367, 378, 379, 382, 383, 406, 407, 413, 415, 431, 443, 446

Offset

1,2

Comments

Each integer k can be mapped onto a black-and-white checkerboard pattern if we read the digits of its binary representation into the diagonals of a quadrant form, the least-significant digit b(0) into the corner, taking 1, then 2, then 3 etc. digits to fill consecutive diagonals of the quadrant:

b(0) b(2) b(5) b(9) ..

b(1) b(4) b(8) ..

b(3) b(7) ..

b(6) ..

b(10)

This will leave one last diagonal partially filled unless the number of binary digits in k is a triangular number.

Replace the "1" bits by black unit squares and the "0" or unset bits by white squares.

If the black squares define a singly connected polyomino considering position and up-down-left-right connectivity, and if the same polyomino cannot be created by a number smaller than k, we add k to the sequence. Polyominoes are considered the same if they can be matched by translations, rotations or flips.

Links

Table of n, a(n) for n=1..56.

Example

(i) The triangular representations of 3= 11, 5 = 101, 10=1010 and are
1
1
and
11
0
and
00
1
1
The 1's define the same 2-omino in all 3 cases, so only the smallest representative, the 3, enters the sequence.
(ii) For k =181 = 10110101 the quadrant is filled with
111
01
01
0
No smaller number leads with this method to this T-shaped 5-omino, so 181 enters the sequence.
(iii) the representations of 6=110, 9= 1001 and 29 =11101 are
01
1
and
10
0
1
and
11
01
1
In all of these 3 cases, the 1's are not singly connected and do not represent polyominoes, so neither 6 nor 9 nor 29 can make it into the sequence.

Crossrefs

Sequence in context: A163094 A022800 A071849 * A246559 A246521 A160785

Adjacent sequences: A165194 A165195 A165196 * A165198 A165199 A165200

Keyword

nonn,base

Author

Leonhard Kreissig, Sep 06 2009

Extensions

Explanation expanded - R. J. Mathar, Sep 22 2009

Status

approved