A007802 -id:A007802 - cp's OEIS frontend

A118141 Length of the longest perfect parity pattern with n columns.

2, 3, 5, 4, 23, 8, 11, 27, 29, 30, 47, 62, 17, 339, 23, 254, 167, 512, 59, 2339, 185, 2046, 95, 1024, 125, 2043, 35, 3276, 2039, 340, 47, 4091, 509, 4094, 335, 3590, 1025, 16379, 119, 1048574, 4679, 16382, 371, 92819, 12281, 8388606, 191, 2097152, 6149, 262139

Offset: 1

Don Knuth, May 11 2006

nonn

Also the length of the unique perfect parity pattern whose first row is 0....01 (with n-1 zeros).
Definitions: A parity pattern is a matrix of 0's and 1's with the property that every 0 is adjacent to an even number of 1's and every 1 is adjacent to an odd number of 1's.
It is called perfect if no row or column is entirely zero. Every parity pattern can be built up in a straightforward way from the smallest perfect subpattern in its upper left corner.
For example, the 3 X 2 matrix
11
00
11
is a parity pattern built up from the perfect 1 X 2 pattern "11". The 3 X 5 matrix
01010
11011
01010
is similarly built up from the perfect 3 X 2 pattern of its first two columns. The 4 X 4 matrix
0011
0100
1101
0101
is perfect. So is the 5 X 5
01110
10101
11011
10101
01110
which moreover has 8-fold symmetry (cf. A118143).
All perfect parity patterns of n columns can be shown to have length d-1 where d divides a(n)+1.

D. E. Knuth, The Art of Computer Programming, Section 7.1.3.

Andries E. Brouwer, Jun 15 2008, Table of n, a(n) for n = 1..85
Andries E. Brouwer, Button Madness and Lights Out on rectangles

The number of perfect parity patterns that have exactly n columns is A000740.
The sequence of all n such that an n X n parity pattern exists is A117870 (cf. A076436, A093614, A094425).
Cf. also A118142, A118143.
Cf. A007802.

More terms from John W. Layman, May 17 2006

More terms from Andries E. Brouwer, Jun 15 2008

A165740 Positive integers n such that solution to the toric n X n "Lights Out" puzzle is not unique (up to the order of flippings; each flipping appears at most once).

Original entry on oeis.org

3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 21, 24, 25, 27, 30, 31, 33, 34, 35, 36, 39, 40, 42, 45, 48, 50, 51, 54, 55, 57, 60, 62, 63, 65, 66, 68, 69, 70, 72, 75, 78, 80, 81, 84, 85, 87, 90, 93, 95, 96, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 119, 120, 123, 124, 125, 126

Offset: 1

Max Alekseyev, Sep 25 2009

nonn

Complement to the sequence A165741 in the set of positive integers.

Any positive multiple of a member of this sequence is also a member. Primitive elements are in A007802. - Thomas Buchholz, May 23 2014

See A165738 for the references.

Max Alekseyev and Thomas Buchholz, Table of n, a(n) for n = 1..1000 [terms 67 through 1000 were computed by Thomas Buchholz, May 20 2014]

Cf. A159257, A075462, A117870, A165741, A007802, A165738.

A number n is in this sequence iff A165738(n) > 0.

More terms from Thomas Buchholz, May 20 2014

cp's OEIS Frontend

A118141 Length of the longest perfect parity pattern with n columns.

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