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%I #18 Apr 28 2017 16:43:47
%S 17,21,29,41,57,77,101,129,161,197,237,281,329,381,437,497,561,629,
%T 701,777,857,941,1029,1121,1217,1317,1421,1529,1641,1757,1877,2001,
%U 2129,2261,2397,2537,2681,2829,2981,3137,3297,3461,3629,3801,3977,4157,4341,4529
%N a(n) = 2*n^2 - 2*n + 17.
%C a(n) is the curvature of the n-th touching circle in the area below the counterclockwise Pappus chain and the left semicircle of the arbelos with radii r0 = 2/3, r1 = 1/3. See illustration in the links.
%H Colin Barker, <a href="/A259555/b259555.txt">Table of n, a(n) for n = 1..1000</a>
%H Kival Ngaokrajang, <a href="/A259555/a259555.pdf">Illustration of initial terms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DescartesCircleTheorem.html">Descartes Circle theorem</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PappusChain.html">Pappus chain</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Descartes%27_theorem">Descartes' Theorem</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 2*n^2 - 2*n + 17.
%F Descartes three circle theorem: a(n) = 3/2 + c(n) + c(n-1) + 2*sqrt(3*(c(n)+ c(n-1)/2 + c(n)*c(n-1)), with c(n) = A114949(n)/2 = (n^2 + 6)/2, producing 2*n^2 - 2*n + 17. - _Wolfdieter Lang_, Jun 30 2015
%F a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - _Colin Barker_, Jul 01 2015
%F G.f.: -x*(17*x^2-30*x+17) / (x-1)^3. - _Colin Barker_, Jul 01 2015
%t Table[2*n^2 - 2*n + 17, {n, 50}] (* _Wesley Ivan Hurt_, Feb 04 2017 *)
%t LinearRecurrence[{3,-3,1},{17,21,29},50] (* _Harvey P. Dale_, Apr 28 2017 *)
%o (PARI) a(n)=2*n^2-2*n+17
%o for (n=1,100,print1(a(n),", "))
%o (PARI) Vec(-x*(17*x^2-30*x+17)/(x-1)^3 + O(x^100)) \\ _Colin Barker_, Jul 01 2015
%Y Cf. A242412 (for r0 = 1/2 = r1), A114949.
%K nonn,easy
%O 1,1
%A _Kival Ngaokrajang_, Jun 30 2015
%E Edited by _Wolfdieter Lang_, Jun 30 2015
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