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A269745 Maximal number of 1's in an n X n {0,1} Toeplitz matrix with property that no four 1's form a square with sides parallel to the edges of the matrix. 2
1, 3, 6, 10, 14, 18, 23, 29, 36, 44, 52, 60, 68, 76 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Label the entries in the left edge and top row (reading from the bottom left to the top right) with the numbers 1 through 2n-1, and let S denote the subset of [1..2n-1] where the matrix contains 1's. Then the matrix has the no-subsquare property iff S contains no three-term arithmetic progression.

LINKS

Table of n, a(n) for n=1..14.

EXAMPLE

n, a(n), example of optimal S:

1, 1, [1]

2, 3, [1, 2]

3, 6, [1, 3, 4]

4, 10, [1, 2, 4, 5]

5, 14, [2, 3, 5, 6]

6, 18, [3, 4, 6, 7]

7, 23, [1, 5, 7, 8, 10]

8, 29, [1, 2, 7, 8, 10, 11]

9, 36, [1, 3, 4, 9, 10, 12, 13]

10, 44, [1, 2, 4, 5, 10, 11, 13, 14]

11, 52, [2, 3, 5, 6, 11, 12, 14, 15]

12, 60, [3, 4, 6, 7, 12, 13, 15, 16]

13, 68, [4, 5, 7, 8, 13, 14, 16, 17]

14, 76, [5, 6, 8, 9, 14, 15, 17, 18]

...

For example, the line 5, 14, [2, 3, 5, 6] corresponds to the Toeplitz matrix

11000

01100

10110

11011

01101

and the value a(5) = 14.

CROSSREFS

This is a lower bound on A227133.

See A269746 for the analogous sequence for a triangular grid.

Cf. A003002.

Sequence in context: A145913 A130246 A167381 * A259646 A036572 A139328

Adjacent sequences:  A269742 A269743 A269744 * A269746 A269747 A269748

KEYWORD

nonn,more

AUTHOR

Warren D. Smith and N. J. A. Sloane, Mar 19 2016

EXTENSIONS

a(14) from N. J. A. Sloane, May 05 2016

STATUS

approved

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Last modified February 21 04:43 EST 2018. Contains 299389 sequences. (Running on oeis4.)