A245302 Curvature (truncated to integer) of a circle inscribed between a unit circle and a vertex of a circumscribed regular n-gon.

3, 5, 9, 13, 19, 25, 32, 39, 48, 57, 67, 78, 90, 103, 116, 130, 145, 161, 178, 195, 213, 232, 252, 273, 294, 317, 340, 364, 388, 414, 440, 467, 495, 524, 554, 584, 615, 647, 680, 714, 748, 783, 820, 856, 894, 933, 972, 1012, 1053, 1095, 1137, 1181, 1225, 1270, 1316, 1362

Offset

3,1

Comments

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

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

Links

Table of n, a(n) for n=3..58.

Kival Ngaokrajang, Illustration of initial terms.

Kival Ngaokrajang, The circles between a unit circle and an inscribed regular n-gon.

Eric Weisstein's World of Mathematics, Incircle.

Formula

a(n) = floor(1/sin(Pi/(2*n))^2 - 1) for n > 2. - Michael Somos, Aug 05 2014

Maple

A245302:=n->floor(sin(Pi/(2*n))^(-2)-1): seq(A245302(n), n=3..100); # Wesley Ivan Hurt, Jul 18 2015

Mathematica

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 *)

Prog

(PARI)
{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, ", "))}
(PARI) {a(n) = if( n<4, 3*(n==3), floor(sin(Pi/2 / n)^-2) - 1)}; /* Michael Somos, Aug 05 2014 */

Crossrefs

Sequence in context: A207187 A359185 A065802 * A118028 A209974 A099392

Adjacent sequences: A245299 A245300 A245301 * A245303 A245304 A245305

Keyword

nonn,easy,changed

Author

Kival Ngaokrajang, Jul 17 2014

Status

approved