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The number of regions into which the plane is divided by a hypotrochoid with parameters R = d = prime(n+1) and r = prime(n).
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%I #20 Jul 20 2023 07:20:18

%S 2,7,9,35,15,53,21,71,147,33,187,125,45,143,267,297,63,337,215,75,397,

%T 251,447,681,305,105,323,111,341,1653,395,687,141,1343,153,787,817,

%U 503,867,897,183,1721,195,593,201,2323,2455,683,231,701,1197

%N The number of regions into which the plane is divided by a hypotrochoid with parameters R = d = prime(n+1) and r = prime(n).

%C The hypotrochoid is given by three parameters: R is the radius of the fixed circle, r is the radius of the rotating disk, d is the distance from the "drawing" point of the disk to the center of this disk.

%H Nicolay Avilov, <a href="/A360734/a360734.jpg">Illustration for terms a(1) - a(10) of the sequence</a>

%F a(n) = prime(n+1)*(prime(n+1) - prime(n) - 1) + 2.

%e See illustration in Links.

%e a(4) = 11*(11 - 7 - 1) + 2 = 35.

%t a[n_]:=Prime[n+1]*(Prime[n+1] - Prime[n]- 1) + 2; Array[a,51] (* _Stefano Spezia_, Feb 18 2023 *)

%Y Cf. A000040.

%K nonn

%O 1,1

%A _Nicolay Avilov_, Feb 18 2023