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 A259555 a(n) = 2*n^2 - 2*n + 17. 5
 17, 21, 29, 41, 57, 77, 101, 129, 161, 197, 237, 281, 329, 381, 437, 497, 561, 629, 701, 777, 857, 941, 1029, 1121, 1217, 1317, 1421, 1529, 1641, 1757, 1877, 2001, 2129, 2261, 2397, 2537, 2681, 2829, 2981, 3137, 3297, 3461, 3629, 3801, 3977, 4157, 4341, 4529 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. LINKS Colin Barker, Table of n, a(n) for n = 1..1000 Kival Ngaokrajang, Illustration of initial terms Eric Weisstein's World of Mathematics, Descartes Circle theorem Eric Weisstein's World of Mathematics, Pappus chain Wikipedia, Descartes' Theorem Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = 2*n^2 - 2*n + 17. 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 a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Colin Barker, Jul 01 2015 G.f.: -x*(17*x^2-30*x+17) / (x-1)^3. - Colin Barker, Jul 01 2015 MATHEMATICA Table[2*n^2 - 2*n + 17, {n, 50}] (* Wesley Ivan Hurt, Feb 04 2017 *) LinearRecurrence[{3, -3, 1}, {17, 21, 29}, 50] (* Harvey P. Dale, Apr 28 2017 *) PROG (PARI) a(n)=2*n^2-2*n+17 for (n=1, 100, print1(a(n), ", ")) (PARI) Vec(-x*(17*x^2-30*x+17)/(x-1)^3 + O(x^100)) \\ Colin Barker, Jul 01 2015 CROSSREFS Cf. A242412 (for r0 = 1/2 = r1), A114949. Sequence in context: A128546 A188200 A060875 * A138600 A050845 A219396 Adjacent sequences:  A259552 A259553 A259554 * A259556 A259557 A259558 KEYWORD nonn,easy AUTHOR Kival Ngaokrajang, Jun 30 2015 EXTENSIONS Edited by Wolfdieter Lang, Jun 30 2015 STATUS approved

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Last modified September 19 04:52 EDT 2019. Contains 327187 sequences. (Running on oeis4.)