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A099938 Consider the sequence of circles C0, C1, C2, C3 ..., where C0 is a half-circle of radius 1. C1 is the largest circle that fits into C0 and has radius 1/2. C(n+1) is the largest circle that fits inside C0 but outside C(n), etc. Sequence gives the curvatures (reciprocals of the radii) of the circles. 2
2, 4, 18, 100, 578, 3364, 19602, 114244, 665858, 3880900, 22619538, 131836324, 768398402, 4478554084, 26102926098, 152139002500, 886731088898, 5168247530884, 30122754096402, 175568277047524, 1023286908188738, 5964153172084900, 34761632124320658 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The numbers a(2), a(4), a(6) etc. are squares and a(1), a(3), a(5) ... are twice squares. Furthermore, a(1) - 2, a(3) - 2, a(5) - 2 etc. are squares and a(2) - 2, a(4) - 2, a(6) - 2 etc. are twice square.

C(n) is centered at (x(n), y(n)), where x(n) = sqrt(1 - 2/a(n)) and y(n) = 1/a(n). - David Wasserman, Feb 28 2008

C(n) is tangent to C0 because sqrt(x(n)^2 + y(n)^2) + y(n) = 1 and C(n) is tangent to C(n+1) because sqrt[(x(n+1) - x(n))^2 + (y(n+1) - y(n))^2] = y(n) + y(n+1). - David Wasserman, Feb 28 2008

a(n+1)/a(n) converges to 3 + sqrt(8). - David Wasserman, Feb 28 2008

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Francisco Javier Garcia Capitan Blog, About the A099938 sequence

David Wasserman, Illustration of this sequence

Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

FORMULA

a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3). G.f.: -2*x*(2*x^2-5*x+1) / ((x-1)*(x^2-6*x+1)). - Colin Barker, Jan 07 2013

a(n) = 1/2*(2 + (3 - 2*sqrt(2))^n*(3 + 2*sqrt(2)) + (3 - 2*sqrt(2))*(3 + 2*sqrt(2))^n). - Colin Barker, Jun 05 2016

MATHEMATICA

Table[FullSimplify[2 Cosh[n ArcSinh[1]]^2], {n, 0, 9}] (* Francisco Javier García Capitán, Jun 05 2016 *)

PROG

(PARI) Vec(-2*x*(2*x^2-5*x+1)/((x-1)*(x^2-6*x+1)) + O(x^30)) \\ Colin Barker, Jun 05 2016

CROSSREFS

Equals 2 * A055997(n-1).

Sequence in context: A120664 A095816 A020101 * A135069 A067647 A208358

Adjacent sequences:  A099935 A099936 A099937 * A099939 A099940 A099941

KEYWORD

nonn,easy

AUTHOR

Hartmut Neubauer (hartmut.f.neubauer(AT)t-online.de), Nov 12 2004

EXTENSIONS

More terms from David Wasserman, Feb 28 2008

STATUS

approved

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Last modified August 6 00:36 EDT 2021. Contains 346493 sequences. (Running on oeis4.)