A243618 Table read by antidiagonals: T(n,k) is the curvature of a circle in a nested Pappus chain (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

Author

Kival Ngaokrajang, Jun 07 2014

Keywords

• nonn
• tabl

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 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.

Examples

			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  ...
..   ..  ..  ..  ...  ...  ...  ...
		

Links

• Kival Ngaokrajang, Illustration of initial terms.
• Wikipedia, Apollonian gasket.
• Wikipedia, Pappus chain.

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).