OFFSET
0,2
COMMENTS
Maximum number of regions that can be formed in the plane by drawing n circles (of any size), also maximum number of regions that can be formed on the sphere by drawing n great circles.
It is unfortunate that A014206 (which should have been this sequence) starts 2, 4, 8, 14, 22, 32, 44, 58, 74, 92, ... and has offset 0, but it is much too late to change it now. A014206 is, however, the main entry for this problem and the present sequence has been created to serve as a pointer to it.
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 3.
Jacob Steiner, Einige Gesetze über die Theilung der Ebene und des Raumes, J. Reine Angew. Math., 1 (1826), 349-364. See Eq. (11). [Note that the title uses the old spelling of Teilung. This is not a typo. - N. J. A. Sloane, Mar 30 2026]
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..10000
David O. H. Cutler, Jonas Karlsson, and Neil J. A. Sloane, Cutting a Pancake with an Exotic Knife, arXiv:2511.15864[math.CO], v3, April 19 2026.
Johan Nilsson, The Pattern Complexity of the Sierpiński Triangle, arXiv:2510.06493 [math.CO], 2025. See p. 2.
N. J. A. Sloane, 5 circles can divide the plane into a(5) = 22 regions [The circles here are all the same size, although that was not a requirement. The same construction works for any n: take n equally-spaced centers around a circle. Then use inverse stereographic progression to get n great circles on a sphere.]
Eric Weisstein's World of Mathematics, Plane Division by Circles.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
From Stefano Spezia, Aug 01 2025: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3.
G.f.: (1 - x + x^2 + x^3)/(1 - x)^3.
E.g.f.: exp(x)*(2 + x^2) - 1. (End)
a(n) = 2*A152947(n+1) for n > 0. - Enrique Navarrete, Oct 16 2025
MATHEMATICA
A386480[n_] := If[n == 0, 1, n*(n - 1) + 2];
Array[A386480, 100, 0] (* Paolo Xausa, Aug 01 2025 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Aug 01 2025
STATUS
approved
