This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A300153 #20 Feb 16 2025 08:33:53 %S A300153 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 A300153 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 A300153 20,6,3,2,4,20,1,21,4,3,5,11,24,44,60,72,80,84,84,80 %N 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)). %C A300153 For any real p > 0, the rose or rhodonea curve with polar coordinates r = cos(t * p): %C A300153 - is dense in the unit disk when p is irrational, %C A300153 - 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 A300153 - see also the illustration in Links section. %H A300153 Rémy Sigrist, <a href="/A300153/a300153.png">Illustration of the first terms</a> %H A300153 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rose.html">Rose</a> %H A300153 Wikipedia, <a href="https://en.wikipedia.org/wiki/Rose_(mathematics)">Rose (mathematics)</a> %F A300153 T(1, k) = A022998(k). %F A300153 T(n, k) = T(n/gcd(n, k), k/gcd(n, k)). %F A300153 Empirically, when gcd(n, k) = 1, we have the following formulas depending on the parity of n and of k: %F A300153 | k is odd | k is even %F A300153 ----------+--------------------------------+-------------------- %F A300153 n is odd | T(n, k) = k * A029578(n+1) | T(n, k) = 2 * k * n %F A300153 n is even | T(n, k) = 2 * k * A029578(n+1) | N/A %e A300153 Array T(n, k) begins: %e A300153 n\k| 1 2 3 4 5 6 7 8 9 %e A300153 ---+--------------------------------------------- %e A300153 1| 1 4 3 8 5 12 7 16 9 %e A300153 2| 4 1 12 4 20 3 28 8 36 %e A300153 3| 2 12 1 24 10 4 14 48 3 %e A300153 4| 8 4 24 1 40 12 56 4 72 %e A300153 5| 3 20 9 40 1 60 21 80 27 %e A300153 6| 12 2 4 12 60 1 84 24 12 %e A300153 7| 4 28 12 56 20 84 1 112 36 %e A300153 8| 16 8 48 4 80 24 112 1 144 %e A300153 9| 5 36 2 72 25 12 35 144 1 %e A300153 10| 20 3 60 20 4 9 140 40 180 %e A300153 11| 6 44 18 88 30 132 42 176 54 %e A300153 ... %e A300153 The following diagram shows the curve for T(2, 1) and the corresponding 4 parts: %e A300153 | %e A300153 ######## ######## %e A300153 ##### ####### ##### %e A300153 ### ### ### ### %e A300153 ### ## | ## ### %e A300153 ## ## ## ## %e A300153 ## # Part #2 # ## %e A300153 ## ## ## ## %e A300153 # ### | ### # %e A300153 -#- - - Part #3 - -#######- - Part #1 - - -#- %e A300153 # ### | ### # %e A300153 ## ## ## ## %e A300153 ## # Part #4 # ## %e A300153 ## ## ## ## %e A300153 ### ## | ## ### %e A300153 ### ### ### ### %e A300153 ##### ####### ##### %e A300153 ######## ######## %e A300153 | %Y A300153 Cf. A022998, A029578. %K A300153 nonn,tabl %O A300153 1,2 %A A300153 _Rémy Sigrist_, Feb 26 2018