A243618 - OEIS

%I #20 May 25 2024 23:47:39

%S 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 29,28,31,38,72,57,46,39,36,37,42,51,90,73,60,51,46,45,48,55,66,110,

%U 91,76,65,58,55,56,61,70,83,132

%N Table read by antidiagonals: T(n,k) is the curvature of a circle in a nested Pappus chain (see Comments for details).

%C Refer to sequential curvatures from Wikipedia. For any integer k > 0, there exists an Apollonian gasket defined by the following curvatures:

%C   (-k, k+1, k*(k+1), k*(k+1)+1).

%C 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 Kival Ngaokrajang, <a href="/A243618/a243618.pdf">Illustration of initial terms</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Apollonian_gasket">Apollonian gasket</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pappus_chain">Pappus chain</a>.

%e Table begins:

%e n/k   1   2   3    4    5    6    7  ...

%e 0     2   6  12   20   30   42   56  ...

%e 1     3   7  13   21   31   43   57  ...

%e 2     6  10  16   24   34   46   60  ...

%e 3    11  15  21   29   39   51   65  ...

%e 4    18  22  28   36   46   58   72  ...

%e 5    27  31  37   45   55   67   80  ...

%e 6    38  42  48   56   66   78   91  ...

%e 7    51  55  61   68   79   91  105  ...

%e 8    66  70  76   83   94  106  120  ...

%e 9    83  87  93  101  111  123  137  ...

%e ..   ..  ..  ..  ...  ...  ...  ...

%o (Small Basic)

%o For k=1 to 50

%o   a=-1*(1/k)

%o   b=1/(k+1)

%o   c=1/(k*(k+1))

%o   aa[0][k]=k*(k+1)

%o   For n = 1 To 50

%o     x=a*b*c

%o     y=Math.Power(x*(a+b+c),1/2)

%o     r=x/(a*b+a*c+b*c-2*y)

%o     aa[n][k]= Math.Round(1/r)

%o     c=r

%o   EndFor

%o EndFor

%o '=====================================

%o For t = 1 to 20

%o   d=0

%o   For nn=0 To t-1

%o     kk=t-d

%o     TextWindow.Write(aa[nn][kk]+", ")

%o     d=d+1

%o   EndFor

%o Endfor

%Y 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 nonn,tabl

%O 0,1

%A _Kival Ngaokrajang_, Jun 07 2014