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 A243618 #20 May 25 2024 23:47:39 %S A243618 2,6,3,12,7,6,20,13,10,11,30,21,16,15,18,42,31,24,21,22,27,56,43,34, %T A243618 29,28,31,38,72,57,46,39,36,37,42,51,90,73,60,51,46,45,48,55,66,110, %U A243618 91,76,65,58,55,56,61,70,83,132 %N A243618 Table read by antidiagonals: T(n,k) is the curvature of a circle in a nested Pappus chain (see Comments for details). %C A243618 Refer to sequential curvatures from Wikipedia. For any integer k > 0, there exists an Apollonian gasket defined by the following curvatures: %C A243618 (-k, k+1, k*(k+1), k*(k+1)+1). %C A243618 For example, the gaskets defined by (-1, 2, 2, 3), (-2, 3, 6, 7), (-3, 4, 12, 13), ..., all follow this pattern (all curvatures are integral). Because every interior circle that is defined by k+1 can become the bounding circle (defined by -k) in another gasket, these gaskets can be nested. When one considers only circles that contact both circles -k and k+1, the pattern will be nested Pappus chains. T(n,k) is the curvature when n = 0 is the circle at the center and n > 0 is in the clockwise direction, k >= 1 for each nested iteration. See illustration in links. %H A243618 Kival Ngaokrajang, <a href="/A243618/a243618.pdf">Illustration of initial terms</a>. %H A243618 Wikipedia, <a href="http://en.wikipedia.org/wiki/Apollonian_gasket">Apollonian gasket</a>. %H A243618 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pappus_chain">Pappus chain</a>. %e A243618 Table begins: %e A243618 n/k 1 2 3 4 5 6 7 ... %e A243618 0 2 6 12 20 30 42 56 ... %e A243618 1 3 7 13 21 31 43 57 ... %e A243618 2 6 10 16 24 34 46 60 ... %e A243618 3 11 15 21 29 39 51 65 ... %e A243618 4 18 22 28 36 46 58 72 ... %e A243618 5 27 31 37 45 55 67 80 ... %e A243618 6 38 42 48 56 66 78 91 ... %e A243618 7 51 55 61 68 79 91 105 ... %e A243618 8 66 70 76 83 94 106 120 ... %e A243618 9 83 87 93 101 111 123 137 ... %e A243618 .. .. .. .. ... ... ... ... %o A243618 (Small Basic) %o A243618 For k=1 to 50 %o A243618 a=-1*(1/k) %o A243618 b=1/(k+1) %o A243618 c=1/(k*(k+1)) %o A243618 aa[0][k]=k*(k+1) %o A243618 For n = 1 To 50 %o A243618 x=a*b*c %o A243618 y=Math.Power(x*(a+b+c),1/2) %o A243618 r=x/(a*b+a*c+b*c-2*y) %o A243618 aa[n][k]= Math.Round(1/r) %o A243618 c=r %o A243618 EndFor %o A243618 EndFor %o A243618 '===================================== %o A243618 For t = 1 to 20 %o A243618 d=0 %o A243618 For nn=0 To t-1 %o A243618 kk=t-d %o A243618 TextWindow.Write(aa[nn][kk]+", ") %o A243618 d=d+1 %o A243618 EndFor %o A243618 Endfor %Y A243618 Cf. Column 1 = A059100(n), column 2 = A114949(n), column 3 = A241748(n), column 4 = A241850(n), column 5 = A114964(n), row 0 = A002378(k), row 1 = A002061(k+1). %K A243618 nonn,tabl %O A243618 0,1 %A A243618 _Kival Ngaokrajang_, Jun 07 2014