A293030 Square array read by antidiagonals downwards: a(n, k) is smallest number m > a(n, k-1) that is not in a previous row, and that avoids any 3-term arithmetic progression in a(n, 1), a(n, 2), ..., a(n, k-1), m.

1, 2, 3, 4, 6, 8, 5, 7, 9, 22, 10, 12, 17, 23, 24, 11, 15, 18, 25, 27, 65, 13, 16, 20, 26, 51, 66, 70, 14, 19, 21, 49, 54, 68, 71, 72, 28, 30, 35, 50, 60, 69, 79, 81, 194, 29, 33, 36, 52, 63, 74, 80, 153, 195, 208, 31, 34, 44, 53, 64, 75, 151, 162, 197, 209

Offset

1,2

Comments

a(n, 1) is the least positive integer not in a previous row.

This sequence is a permutation of the natural numbers.

The first row is A003278.

The second row is the lexicographically earliest sequence of terms not in the first row that contains no arithmetic progressions.

The third row is the lexicographically earliest sequence of terms not in the first two rows that contains no arithmetic progressions.

Links

Peter Kagey, First 90 rows, flattened

Example

a(2, 4) = 12 because row 2 begins (3, 6, 7)
a(2, 4) != 8 because (6, 7, 8) is an arithmetic progression,
a(2, 4) != 9 because (3, 6, 9) is an arithmetic progression,
a(2, 4) != 10 because a(1, 5) = 10 is in a previous row.
a(2, 4) != 11 because a(1, 6) = 11 is in a previous row.
a(2, 4) = 12 does not create any arithmetic progressions in row 2, and 12 is not in row 1.
Array begins:
    1   2   4   5  10  11  13  14  28  29 ...
    3   6   7  12  15  16  19  30  33  34 ...
    8   9  17  18  20  21  35  36  44  45 ...
   22  23  25  26  49  50  52  53  58  59 ...
   24  27  51  54  60  63  64  67  73  76 ...
   65  66  68  69  74  75  77  78 146 147 ...
   70  71  79  80 151 152 160 161 178 179 ...
   72  81 153 162 180 189 192 193 196 201 ...
  194 195 197 198 203 204 206 207 221 222 ...
  208 209 211 212 235 236 238 239 451 452 ...
  ...

Crossrefs

Cf. A003278.

Sequence in context: A339361 A166310 A385590 * A109852 A083197 A235262

Adjacent sequences: A293027 A293028 A293029 * A293031 A293032 A293033

Keyword

nonn,tabl,look

Author

Peter Kagey, Sep 28 2017

Status

approved