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A243618
Table read by antidiagonals: T(n,k) is the curvature of a circle in a nested Pappus chain (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
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 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.
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
KEYWORD
nonn,tabl
AUTHOR
Kival Ngaokrajang, Jun 07 2014
STATUS
approved