This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A099938 #27 Jun 10 2016 13:47:18
%S A099938 2,4,18,100,578,3364,19602,114244,665858,3880900,22619538,131836324,
%T A099938 768398402,4478554084,26102926098,152139002500,886731088898,
%U A099938 5168247530884,30122754096402,175568277047524,1023286908188738,5964153172084900,34761632124320658
%N 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.
%C A099938 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 A099938 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 A099938 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
%C A099938 a(n+1)/a(n) converges to 3 + sqrt(8). - _David Wasserman_, Feb 28 2008
%H A099938 Colin Barker, <a href="/A099938/b099938.txt">Table of n, a(n) for n = 1..1000</a>
%H A099938 Francisco Javier Garcia Capitan Blog, <a href="http://garciacapitan.blogspot.com.es/2016/06/about-a099938-sequence.html">About the A099938 sequence</a>
%H A099938 David Wasserman, <a href="/A099938/a099938.jpg">Illustration of this sequence</a>
%H A099938 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-7,1).
%F A099938 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
%F A099938 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
%t A099938 Table[FullSimplify[2 Cosh[n ArcSinh[1]]^2], {n, 0, 9}] (* _Francisco Javier García Capitán_, Jun 05 2016 *)
%o A099938 (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
%Y A099938 Equals 2 * A055997(n-1).
%K A099938 nonn,easy
%O A099938 1,1
%A A099938 Hartmut Neubauer (hartmut.f.neubauer(AT)t-online.de), Nov 12 2004
%E A099938 More terms from _David Wasserman_, Feb 28 2008