A345067 Consider the "Quilt Tiling"; T(n, k) is the area of the tile containing the unit square whose upper right corner has coordinates (n, k); square array T(n, k) read by antidiagonals upwards, n, k > 0.

1, 2, 2, 2, 4, 2, 6, 4, 4, 6, 6, 6, 4, 6, 6, 6, 6, 1, 1, 6, 6, 15, 6, 2, 9, 2, 6, 15, 15, 15, 2, 9, 9, 2, 15, 15, 15, 15, 15, 9, 9, 9, 15, 15, 15, 15, 15, 15, 2, 9, 9, 2, 15, 15, 15, 15, 15, 15, 2, 4, 9, 4, 2, 15, 15, 15, 40, 15, 15, 6, 4, 4, 4, 4, 6, 15, 15, 40

Offset: 1

Author

Rémy Sigrist, Jun 06 2021

Keywords

• nonn
• look
• tabl

Comments

The "Quilt Tiling" is described in Shectman's paper (see Links section).

All terms belong to A006498.

Examples

			Array T(n, k) begins:
  n\k|  1   2   3   4   5   6   7   8   9  10  11
  ---+---+-------+-----------+-------------------+
   1 |  1|  2   2|  6   6   6| 15  15  15  15  15|
     +-----------+           |                   |
   2 |  2|  4   4|  6   6   6| 15  15  15  15  15|
     |   |       +---+-------+                   |
   3 |  2|  4   4|  1|  2   2| 15  15  15  15  15|
     +---+---+---+---+-------+-------+-----------+
   4 |  6   6|  1|  9   9   9|  2   2|  6   6   6|
     |       +---+           +-------+           |
   5 |  6   6|  2|  9   9   9|  4   4|  6   6   6|
     |       |   |           |       +---+-------+
   6 |  6   6|  2|  9   9   9|  4   4|  1|  2   2|
     +-------+---+---+-------+-------+---+-------+
   7 | 15  15  15|  2|  4   4| 25  25  25  25  25|
     |           |   |       |                   |
   8 | 15  15  15|  2|  4   4| 25  25  25  25  25|
     |           +---+---+---+                   |
   9 | 15  15  15|  6   6|  1| 25  25  25  25  25|
     |           |       +---+                   |
  10 | 15  15  15|  6   6|  2| 25  25  25  25  25|
     |           |       |   |                   |
  11 | 15  15  15|  6   6|  2| 25  25  25  25  25|
     +-----------+-------+---+-------------------+
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10153
• J. Parker Shectman, A Quilt after Fibonacci, Part 2 of 3: Cohorts, Free Monoids, and Numeration
• Rémy Sigrist, Illustration of the connection between the "Quilt Tiling" and the sequences A000201 and A005206
• Rémy Sigrist, PARI program for A345067

Crossrefs

Cf. A000045, A001654, A005206, A006498, A095791, A005206.

Programs

PARI

See Links section.

Formula

T(n, k) = T(k, n).

T(n, n) = A130312(n+1)^2.

T(n, 1) = A001654(A095791(n)+1).

T(n, k) is the square of a Fibonacci number for n = 1+A005206(k+1)..A000201(k).