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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 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 * A310065 A310066 A310067 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 January 19 21:52 EST 2019. Contains 319310 sequences. (Running on oeis4.)