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A165197 Smallest integers whose binary digits "1" define a polyomino if sorted into a quadrant shape. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 16 06:48 EDT 2019. Contains 327090 sequences. (Running on oeis4.)