A300153 Square array T(n, k) read by antidiagonals upwards, n > 0 and k > 0: T(n, k) is the number of parts inscribed in a rose or rhodonea curve with polar coordinates r = cos(t * (k/n)).

1, 4, 4, 2, 1, 3, 8, 12, 12, 8, 3, 4, 1, 4, 5, 12, 20, 24, 24, 20, 12, 4, 2, 9, 1, 10, 3, 7, 16, 28, 4, 40, 40, 4, 28, 16, 5, 8, 12, 12, 1, 12, 14, 8, 9, 20, 36, 48, 56, 60, 60, 56, 48, 36, 20, 6, 3, 2, 4, 20, 1, 21, 4, 3, 5, 11, 24, 44, 60, 72, 80, 84, 84, 80

Offset: 1

Author

Rémy Sigrist, Feb 26 2018

Keywords

• nonn
• tabl

Comments

For any real p > 0, the rose or rhodonea curve with polar coordinates r = cos(t * p):

- is dense in the unit disk when p is irrational,

- is closed when p is rational, say p = u/v in reduced form; in that case, the number of parts inscribed in the curve is T(v, u),

- see also the illustration in Links section.

Examples

			Array T(n, k) begins:
  n\k|    1    2    3    4    5    6    7    8    9
  ---+---------------------------------------------
    1|    1    4    3    8    5   12    7   16    9
    2|    4    1   12    4   20    3   28    8   36
    3|    2   12    1   24   10    4   14   48    3
    4|    8    4   24    1   40   12   56    4   72
    5|    3   20    9   40    1   60   21   80   27
    6|   12    2    4   12   60    1   84   24   12
    7|    4   28   12   56   20   84    1  112   36
    8|   16    8   48    4   80   24  112    1  144
    9|    5   36    2   72   25   12   35  144    1
   10|   20    3   60   20    4    9  140   40  180
   11|    6   44   18   88   30  132   42  176   54
...
The following diagram shows the curve for T(2, 1) and the corresponding 4 parts:
                         |
               ########     ########
           #####      #######      #####
        ###          ###   ###          ###
      ###           ##   |   ##           ###
     ##            ##         ##            ##
    ##             #  Part #2  #             ##
   ##              ##         ##              ##
   #                ###  |  ###                #
  -#- - - Part #3  - -#######- -  Part #1 - - -#-
   #                ###  |  ###                #
   ##              ##         ##              ##
    ##             #  Part #4  #             ##
     ##            ##         ##            ##
      ###           ##   |   ##           ###
        ###          ###   ###          ###
           #####      #######      #####
               ########     ########
                         |
		

Links

• Rémy Sigrist, Illustration of the first terms
• Eric Weisstein's World of Mathematics, Rose
• Wikipedia, Rose (mathematics)

Crossrefs

Cf. A022998, A029578.

Formula

T(1, k) = A022998(k).

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

Empirically, when gcd(n, k) = 1, we have the following formulas depending on the parity of n and of k:

| k is odd | k is even

----------+--------------------------------+--------------------

n is odd | T(n, k) = k * A029578(n+1) | T(n, k) = 2 * k * n

n is even | T(n, k) = 2 * k * A029578(n+1) | N/A