A243618 Table read by antidiagonals: T(n,k) is circle curvature in nested Pappus chains (see Comments for details).

2, 6, 3, 12, 7, 6, 20, 13, 10, 11, 30, 21, 16, 15, 18, 42, 31, 24, 21, 22, 27, 56, 43, 34, 29, 28, 31, 38, 72, 57, 46, 39, 36, 37, 42, 51, 90, 73, 60, 51, 46, 45, 48, 55, 66, 110, 91, 76, 65, 58, 55, 56, 61, 70, 83, 132

Offset

0,1

Comments

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

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

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 circle curvature when n = 0 is the circle at center and n > 0 is in the clockwise direction, k >= 1 for each nested iteration. See illustration in links.

Links

Table of n, a(n) for n=0..55.

Kival Ngaokrajang, Illustration of initial terms

Wikipedia, Apollonian gasket

Wikipedia, Pappus chain

Example

Table begins:
n/k   1   2   3    4    5    6    7  ...
0     2   6  12   20   30   42   56  ...
1     3   7  13   21   31   43   57  ...
2     6  10  16   24   34   46   60  ...
3    11  15  21   29   39   51   65  ...
4    18  22  28   36   46   58   72  ...
5    27  31  37   45   55   67   80  ...
6    38  42  48   56   66   78   91  ...
7    51  55  61   68   79   91  105  ...
8    66  70  76   83   94  106  120  ...
9    83  87  93  101  111  123  137  ...
..   ..  ..  ..  ...  ...  ...  ...

Prog

(Small Basic)
For k=1 to 50
  a=-1*(1/k)
  b=1/(k+1)
  c=1/(k*(k+1))
  aa[0][k]=k*(k+1)
  For n = 1 To 50
    x=a*b*c
    y=Math.Power(x*(a+b+c), 1/2)
    r=x/(a*b+a*c+b*c-2*y)
    aa[n][k]= Math.Round(1/r)
    c=r
  EndFor
EndFor
'=====================================
For t = 1 to 20
  d=0
  For nn=0 To t-1
    kk=t-d
    TextWindow.Write(aa[nn][kk]+", ")
    d=d+1
  EndFor
Endfor

Crossrefs

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).

Sequence in context: A303751 A304531 A304755 * A063929 A276158 A092393

Adjacent sequences: A243615 A243616 A243617 * A243619 A243620 A243621

Keyword

nonn,tabl

Author

Kival Ngaokrajang, Jun 07 2014

Status

approved