A243813 Table read by antidiagonals: T(n,k) is the curvature (truncated to integer) of a circle in a variation of nested Pappus chains (see Comments for details).

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 9, 1, 1, 1, 1, 3, 13, 1, 1, 1, 1, 2, 5, 19, 1, 1, 1, 1, 1, 3, 7, 25, 1, 1, 1, 1, 1, 2, 4, 9, 33, 1, 1, 1, 1, 1, 1, 2, 5, 11, 41, 1, 1, 1, 1, 1, 1, 2, 3, 6, 14, 51, 1, 1, 1, 1, 1, 1, 1, 2, 4, 7, 17, 61, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 9, 21

Offset

0,6

Comments

Refer to the construction rule used in A243618. For this case, the curvature is defined by (-1/k, 1/(k-1), 1), the circle radius will diverge to infinity (zero curvature). The integral curvatures appearing as periodic, i.e., 2, 6, 6, 10, 30, 42, 28, 12, ..., = A083482(k-1). The integral curvatures seem to align as some sequence, e.g., 3, 7, 13, 21, 31, 43, ..., = A002061(k) and 9, 25, 49, ..., = A016754(k-1). See illustration.

Links

Table of n, a(n) for n=0..89.

Kival Ngaokrajang, Illustration for k = 2..7

Example

Table begins:
  n/k  2   3   4   5   6   7  ...
   0   1   1   1   1   1   1  ...
   1   1   1   1   1   1   1  ...
   2   3   1   1   1   1   1  ...
   3   5   2   1   1   1   1  ...
   4   9   3   2   1   1   1  ...
   5  13   5   3   2   1   1  ...
   6  19   7   4   2   2   1  ...
   7  25   9   5   3   2   2  ...
   8  33  11   6   4   3   2  ...
   9  41  14   7   5   3   2  ...
  10  51  17   9   6   4   3  ...
  11  61  21  11   7   5   3  ...
  12  73  25  13   8   5   4  ...
  ...

Prog

(Small Basic)
For k=2 to 100
  a=-k
  b=k-1
  c=1
  aa[0][k]=1
  For n = 1 To 100
    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.floor(1/r)
    If 1/r-math.Floor(1/r)> 0.999999 Then
      aa[n][k]=aa[n][k]+1
    EndIf
    c=r
  EndFor
endFor
'=====================================
For t = 1 to 30
  d=0
  For nn=0 To t-1
    kk=t+1-d
    TextWindow.Write(aa[nn][kk]+", ")
    d=d+1
  EndFor
Endfor

Crossrefs

Cf. A243618, A083482, A002061, A016754.

Cf. Column 1 = A080827(n), column 2 = A056827(n) + 1.

Cf. Integral curvature in column 1..6: [A058331, A227776, A056107, A212656, A158558, A158604].

Sequence in context: A268032 A339899 A305803 * A318453 A318313 A101021

Adjacent sequences: A243810 A243811 A243812 * A243814 A243815 A243816

Keyword

nonn,tabl

Author

Kival Ngaokrajang, Jun 11 2014

Status

approved