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%I #25 Mar 17 2024 03:12:55
%S 3,5,9,13,19,25,32,39,48,57,67,78,90,103,116,130,145,161,178,195,213,
%T 232,252,273,294,317,340,364,388,414,440,467,495,524,554,584,615,647,
%U 680,714,748,783,820,856,894,933,972,1012,1053,1095,1137,1181,1225,1270,1316,1362
%N Curvature (truncated to integer) of a circle inscribed between a unit circle and a vertex of a circumscribed regular n-gon.
%C The curvature is the reciprocal of the radius of any one of the n circles externally tangent to the unit circle and internally tangent to two consecutive sides of the circumscribed regular n-gon. - _Michael Somos_, Aug 05 2014
%C a(n) + 1 is the curvature (truncated to integer) of a circle inscribed between a unit circle and an inscribed regular n-gon. - _Kival Ngaokrajang_, Jul 08 2015
%H Kival Ngaokrajang, <a href="/A245302/a245302.pdf">Illustration of initial terms</a>.
%H Kival Ngaokrajang, <a href="/A245302/a245302_1.pdf">The circles between a unit circle and an inscribed regular n-gon</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Incircle.html">Incircle</a>.
%F a(n) = floor(1/sin(Pi/(2*n))^2 - 1) for n > 2. - _Michael Somos_, Aug 05 2014
%p A245302:=n->floor(sin(Pi/(2*n))^(-2)-1): seq(A245302(n), n=3..100); # _Wesley Ivan Hurt_, Jul 18 2015
%t f[n_] := Block[{a, b, c, d, g, h}, a = Pi(n - 2)/(2n); b = a/2 + Pi/4; c = 1/Tan[a]; d = 1/Tan[b]; g = 2 Tan[b - a]; h = (2c - 2d + g)/2; Floor[1/Sqrt[((h - c + d)^2*(h - g))/h]]]; f[3] = 3; Array[f, 60, 3] (* _Robert G. Wilson v_, Jul 25 2014 *)
%o (PARI)
%o {for (n=3, 100,x1=Pi*(n-2)/(2*n);x2=x1/2+Pi/4;b1=1/tan(x1);b2=1/tan(x2);a=b1-b2;z=x2-x1;c=2*tan(z); s=(2*a+c)/2;r=sqrt(((s-a)^2*(s-c))/s); an=floor(1/r);print1(an,", "))}
%o (PARI) {a(n) = if( n<4, 3*(n==3), floor(sin(Pi/2 / n)^-2) - 1)}; /* _Michael Somos_, Aug 05 2014 */
%K nonn,easy
%O 3,1
%A _Kival Ngaokrajang_, Jul 17 2014
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