A372341 Let F be the set of lattice points {(x, y) in N^2 | A005206(x) <= y <= A005206(x) + x}; order the points of F by ascending Y-coordinates and then by ascending X-coordinates; the n-th and a(n)-th points of F are arranged symmetrically with respect to the line x = y.

1, 2, 4, 3, 5, 7, 6, 8, 11, 16, 9, 12, 17, 22, 29, 10, 13, 18, 23, 30, 37, 14, 19, 24, 31, 38, 46, 56, 15, 20, 25, 32, 39, 47, 57, 67, 21, 26, 33, 40, 48, 58, 68, 79, 92, 27, 34, 41, 49, 59, 69, 80, 93, 106, 121, 28, 35, 42, 50, 60, 70, 81, 94, 107, 122, 137

Offset: 1

Author

Rémy Sigrist, Apr 28 2024

Keywords

• nonn

Comments

The set F is related to the "Quilt Tiling" described in Shectman's paper (see Links section) and has interesting properties: F is symmetrical with respect to the line x = y, for any n >= 0, there are n+1 points in F with a X-coordinate of n (or with a Y-coordinate of n).

This sequence is a self-inverse permutation of the positive integers with infinitely many fixed points (see A372231).

Examples

			The elements of F with coordinates <= 10 are as follows:
     |                       +-------------------+
  10 |                       | 56  57  58  59  60|
     |                       |                   |
   9 |                       | 46  47  48  49  50|
     |                   +---+                   |
   8 |                   | 37| 38  39  40  41  42|
     |               +---+---+                   |
   7 |               | 29  30| 31  32  33  34  35|
     |               |       |                   |
   6 |               | 22  23| 24  25  26  27  28|
     |           +---+-------+-------+---+-------+
   5 |           | 16  17  18| 19  20| 21|
     |           |           |       +---+
   4 |           | 11  12  13| 14  15|
     |       +---+           +-------+
   3 |       |  7|  8   9  10|
     |   +---+---+---+-------+
   2 |   |  4   5|  6|
     |   |       +---+
   1 |   |  2   3|
     +---+-------+
   0 |  1|
  ---+---+----------------------------------------
  y/x|  0   1   2   3   4   5   6   7   8   9  10
So a(1) = 1, a(2) = 2, a(3) = 4, a(5) = 5, a(6) = 7, a(8) = 8, a(9) = 11, a(10) = 16, a(12) = 12, a(13) = 17, etc.
		

Links

• J. Parker Shectman, A Quilt after Fibonacci, Part 2 of 3: Cohorts, Free Monoids, and Numeration
• Rémy Sigrist, C++ program
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A005206, A345067, A372231 (fixed points).