# An Integer Sequence

Antreas P. Hatzipolakis xpolakis at otenet.gr
Sun Apr 30 02:00:13 CEST 2000

```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, ......

In EIS there is the sequence: 1, 5, 13, 25, 41, ......

ID Number: A001844 (Formerly M3826 and N1567)
Sequence:  1,5,13,25,41,61,85,113,145,181,221,265,313,365,421,481,545,
613,685,761,841,925,1013,1105,1201,1301,1405,1513,1625,1741,
1861,1985,2113,2245,2381,2521,2665,2813,2965,3121,3281,3445,
3613,3785,3961,4141,4325
Name:      Centered square numbers: 2n(n+1)+1. Also, consider all Pythagorean
triples (X,Y,Z=Y+1) ordered by increasing Z, sequence gives Z values.
[...]

Antreas

```