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A243618 Table read by antidiagonals: T(n,k) is circle curvature in nested Pappus chains (see Comments for details). 2
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified June 23 07:00 EDT 2021. Contains 345395 sequences. (Running on oeis4.)