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

%I A386480 #27 Aug 03 2025 12:00:48
%S A386480 1,2,4,8,14,22,32,44,58,74,92,112,134,158,184,212,242,274,308,344,382,
%T A386480 422,464,508,554,602,652,704,758,814,872,932,994,1058,1124,1192,1262,
%U A386480 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
%N A386480 a(0) = 1; thereafter a(n) = n^2 - n + 2.
%C A386480 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.
%C A386480 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.
%D A386480 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 73, Problem 3.
%H A386480 Paolo Xausa, <a href="/A386480/b386480.txt">Table of n, a(n) for n = 0..10000</a>
%H A386480 N. J. A. Sloane, <a href="/A386480/a386480.png">5 circles can divide the plane into a(5) = 22 regions</a> [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.]
%H A386480 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PlaneDivisionbyCircles.html">Plane Division by Circles</a>.
%H A386480 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A386480 From _Stefano Spezia_, Aug 01 2025: (Start)
%F A386480 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3.
%F A386480 G.f.: (1 - x + x^2 + x^3)/(1 - x)^3.
%F A386480 E.g.f.: exp(x)*(2 + x^2) - 1. (End)
%F A386480 a(n) = A003682(n) = A002061(n)+1, n>=2. - _R. J. Mathar_, Aug 03 2025
%t A386480 A386480[n_] := If[n == 0, 1, n*(n - 1) + 2];
%t A386480 Array[A386480, 100, 0] (* _Paolo Xausa_, Aug 01 2025 *)
%Y A386480 See A014206 for further information (including additional references).
%K A386480 nonn,easy
%O A386480 0,2
%A A386480 _N. J. A. Sloane_, Aug 01 2025