An Integer Sequence

[Antreas P. Hatzipolakis]

I wrote:

>In the journal _Parabola_ (v. 20, no. 2, 1984, p. 27, #Q607) appeared the
>following problem:
>
>  A number of ellipses are drawn in the plane, any two of them intersecting
>  in 4 points. No three of the curves are concurrent. Three such ellipses
>  divide the plane into 14 regions (including the unbounded region lying
>  outside of all the ellipses). Into how many regions would the plane be
>  divided if 10 ellipses were drawn?
>
>For n ellipses the formula is:  2(n^2 - n + 1)
>So, we have the integer sequence: 2, 6, 14, 26, 42, ......


If any two ellipses intersect in two (instead of four) points, [this is case
the case of circles; see the problems below] and no three are concurrent,
then the formula is:

                          n^2 - n + 2

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In the plane, n circles are drawn so that every two distinct circles meet
in exactly two points and no three of the circles have a common point
Give a formula for the number of regions into which the circles partition
the plane.
(Indiana School Mathematics Journal, vol. 14, no. 4, 1979, p. 4)

One circle divides the plane into 2 regions; two distinct circles give
three or 4 regions, depending on their relative position. Three circles can
yield 8 regions, but not more. Find a formula for the maximum number
of regions obtainable from n circles.

(Parabola, vol. 24, no. 1, 1988, 22, #Q736)
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In EIS:
ID Number: A014206
Sequence:  2,4,8,14,22,32,44,58,74,92,112,134,158,184,212,242,274,308,
           344,382,422,464,508,554,602,652,704,758,814,872,932,994,
           1058,1124,1192,1262,1334,1408,1484,1562,1642,1724
Name:      n^2+n+2.
References J D E Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996,
p. 177.

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Antreas