A358532 a(n) is the row position of the next open point in the structure generated by adding the largest diamond possible at the next open point on a triangular grid of side n. See Comments and Example sections for more details.

1, 1, 2, 1, 2, 3, 1, 4, 1, 3, 7, 1, 3, 6, 4, 10, 1, 9, 4, 7, 9, 5, 14, 1, 11, 5, 7, 8, 11, 14, 19, 1, 6, 6, 24, 9, 14, 20, 1, 8, 8, 8, 20, 8, 19, 24, 30, 15, 19, 19, 19, 27, 1, 19, 15, 16, 20, 28, 8, 39, 11, 24, 1, 11, 16, 26, 28, 29, 30, 39, 50, 20, 31, 32, 33

Offset: 1

Author

John Tyler Rascoe, Nov 20 2022

Keywords

• nonn
• look
• easy

Comments

A structure of diamonds is built up successively by adding the largest possible diamond to the next open point within a triangular grid of side n. Each new diamond is added to the preceding structure of diamonds. At each step n, a new row of n open points is first added, extending the triangular grid.

Then the next open point is defined as the first open point encountered when the triangle is read by rows starting from the top row. a(n) is then the row position of the next open point.

Finally, starting at this open point the largest diamond that does not overlap any previous diamonds and fits within the triangular grid is added. Each diamond of side length k must cover exactly k^2 points, with the top corner on an open point. The points covered by the added diamond are then considered closed.

Is there a pattern for the values of n where a(n) = 1?

Examples

			Here zeros are the open points; closed points covered by the n-th diamond are replaced with n.
  ---------------------
  n=4       1          First a new row of 4 open points is added.
           2 3         Then the next open point is T(3,1) so a(4) = 1.
          4 0 0        Finally, the largest diamond fitting at T(3,1) is 1.
         0 0 0 0
  ---------------------
  n=5       1          First a new row of 5 open points is added.
           2 3         Then the next open point is T(3,2) so a(5) = 2.
          4 5 0        Finally, the largest diamond fitting at T(3,2) is 2.
         0 5 5 0
        0 0 5 0 0
  ---------------------
  n=6       1          First a new row of 6 open points is added.
           2 3         Then the next open point is T(3,3) so a(6) = 3.
          4 5 6        Finally, the largest diamond fitting at T(3,3) is 1.
         0 5 5 0
        0 0 5 0 0
       0 0 0 0 0 0
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 1..10000
• John Tyler Rascoe, Python program

Crossrefs

Cf. A045996, A186434, A194136, A239567, A279413.

Programs

Python

# see linked program