A386480 a(0) = 1; thereafter a(n) = n^2 - n + 2.

1, 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, 1808, 1894, 1982, 2072, 2164, 2258, 2354, 2452, 2552, 2654, 2758, 2864, 2972, 3082, 3194, 3308, 3424, 3542, 3662, 3784

Offset: 0

Author

N. J. A. Sloane, Aug 01 2025

Keywords

• nonn
• easy

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.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• 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).

Crossrefs

See A014206 for further information (including additional references).

Programs

Mathematica

A386480[n_] := If[n == 0, 1, n*(n - 1) + 2];
Array[A386480, 100, 0] (* Paolo Xausa, Aug 01 2025 *)

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) = A003682(n) = A002061(n)+1, n>=2. - R. J. Mathar, Aug 03 2025