References for: A000124,A014206,A000125,A046127

[Antreas P. Hatzipolakis]

Problem:
What is the greatest number of parts into which a plane can be divided by:
a. n straight lines?
b. n circles?

Answer:
a. Formula: (n^2 + n + 2)/2
   Sequence: A000124 = 1,2,4,7,11,16,22,29,37,46,56,67,79,92,106,121,....

b. Formula: n^2 - n + 2
   Sequence: A014206 = 2,4,8,14,22,32,44,58,74,92,112,134,158,184,212,.....

Reference: A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with
           Elementary Solutions. Vol. I. Combinatorial Analysis and
           Probability Theory. New York: Dover Publications, Inc., 1987, p. 13,
           #44 (1st. publ.: San Francisco: Holden-Day, Inc., 1964)

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Problem:
What is the greatest number of parts into which three-dimensional space can
be divided by:
a. n planes?
b. n spheres?

Answer:
a. Formula: (n^3 + 5n + 6)/6
   Sequence: A000125 = 1,2,4,8,15,26,42,64,93,130,176,232,299,....

b. Formula: n*(n^2 - 3n + 8)/3
   Sequence: A046127 = 2,4,8,16,30,52,84,128,186,260,352,464,598,...

Reference: A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with
           Elementary Solutions. Vol. I. Combinatorial Analysis and
           Probability Theory. New York: Dover Publications, Inc., 1987, p. 13,
           #45 (1st. publ.: San Francisco: Holden-Day, Inc., 1964)

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Antreas