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%I #19 Feb 28 2018 08:33:51
%S 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,
%T 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,
%U 20,6,3,2,4,20,1,21,4,3,5,11,24,44,60,72,80,84,84,80
%N 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)).
%C For any real p > 0, the rose or rhodonea curve with polar coordinates r = cos(t * p):
%C - is dense in the unit disk when p is irrational,
%C - 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),
%C - see also the illustration in Links section.
%H Rémy Sigrist, <a href="/A300153/a300153.png">Illustration of the first terms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rose.html">Rose</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Rose_(mathematics)">Rose (mathematics)</a>
%F T(1, k) = A022998(k).
%F T(n, k) = T(n/gcd(n, k), k/gcd(n, k)).
%F Empirically, when gcd(n, k) = 1, we have the following formulas depending on the parity of n and of k:
%F | k is odd | k is even
%F ----------+--------------------------------+--------------------
%F n is odd | T(n, k) = k * A029578(n+1) | T(n, k) = 2 * k * n
%F n is even | T(n, k) = 2 * k * A029578(n+1) | N/A
%e Array T(n, k) begins:
%e n\k| 1 2 3 4 5 6 7 8 9
%e ---+---------------------------------------------
%e 1| 1 4 3 8 5 12 7 16 9
%e 2| 4 1 12 4 20 3 28 8 36
%e 3| 2 12 1 24 10 4 14 48 3
%e 4| 8 4 24 1 40 12 56 4 72
%e 5| 3 20 9 40 1 60 21 80 27
%e 6| 12 2 4 12 60 1 84 24 12
%e 7| 4 28 12 56 20 84 1 112 36
%e 8| 16 8 48 4 80 24 112 1 144
%e 9| 5 36 2 72 25 12 35 144 1
%e 10| 20 3 60 20 4 9 140 40 180
%e 11| 6 44 18 88 30 132 42 176 54
%e ...
%e The following diagram shows the curve for T(2, 1) and the corresponding 4 parts:
%e |
%e ######## ########
%e ##### ####### #####
%e ### ### ### ###
%e ### ## | ## ###
%e ## ## ## ##
%e ## # Part #2 # ##
%e ## ## ## ##
%e # ### | ### #
%e -#- - - Part #3 - -#######- - Part #1 - - -#-
%e # ### | ### #
%e ## ## ## ##
%e ## # Part #4 # ##
%e ## ## ## ##
%e ### ## | ## ###
%e ### ### ### ###
%e ##### ####### #####
%e ######## ########
%e |
%Y Cf. A022998, A029578.
%K nonn,tabl
%O 1,2
%A _Rémy Sigrist_, Feb 26 2018
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