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%I #59 Feb 22 2023 09:13:46
%S 15,23,39,63,95,135,183,239,303,375,455,543,639,743,855,975,1103,1239,
%T 1383,1535,1695,1863,2039,2223,2415,2615,2823,3039,3263,3495,3735,
%U 3983,4239,4503,4775,5055,5343,5639,5943,6255,6575,6903,7239,7583,7935,8295,8663,9039,9423,9815
%N a(n) = (2n-1)^2 + 14.
%C The previous definition was "a(n) = normalized inverse radius of the inscribed circle that is tangent to the left circle of the symmetric arbelos and the n-th and (n-1)-st circles in the Pappus chain".
%C See links section for image of these circles, via Wolfram MathWorld (there an asymmetric arbelos is shown).
%C The Rothman-Fukagawa article has another picture of the circles, based on a Japanese 1788 sangaku problem. - _N. J. A. Sloane_, Jan 02 2020
%H N. J. A. Sloane, <a href="/A242412/b242412.txt">Table of n, a(n) for n = 1..1000</a>
%H Brady Haran and Simon Pampena, <a href="https://www.youtube.com/watch?v=sG_6nlMZ8f4">Epic Circles</a>, Numberphile video (2014).
%H Tony Rothman and Hidetoshi Fukagawa, <a href="https://www.jstor.org/stable/26057787">Japanese temple geometry</a>, Scientific American, Vol. 278, No. 5, May 1998, 85-91.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/images/eps-gif/PappusTangentChain_800.gif">Image of inscribed circles (in red)</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PappusChain.html">Pappus Chain</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pappus_chain">Pappus chain</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 4*n^2 - 4*n + 15.
%F a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). G.f.: -x*(15*x^2-22*x+15) / (x-1)^3. - _Colin Barker_, May 14 2014
%F From Descartes three circle theorem:
%F a(n) = 2 +c(n) + c(n-1) + 2*sqrt(2*(c(n) + c(n-1) + c(n)*c(n-1)), with c(n) = A059100(n) = n^2 +2, n >= 1, which produces 4*n^2 - 4*n + 15. - _Wolfdieter Lang_, Jul 01 2015
%e For n = 1, the radius of the outermost circle divided by the radius of a circle drawn tangent to all three of the initial inner circle, the opposite inner circle (the 0th circle in the chain), and the 1st circle in the chain is 15.
%e For n = 2, the radius of the outermost circle divided by the radius of a circle drawn tangent to all three of the initial inner circle, the 1st circle in the chain, and the 2nd circle in the chain is 23.
%p A242412:=n->4*n^2 - 4*n + 15; seq(A242412(n), n=1..50); # _Wesley Ivan Hurt_, May 13 2014
%t Table[4 n^2 - 4 n + 15, {n, 50}] (* _Wesley Ivan Hurt_, May 13 2014 *)
%t LinearRecurrence[{3,-3,1},{15,23,39},50] (* _Harvey P. Dale_, Feb 22 2023 *)
%o (Magma) [4*n^2 - 4*n + 15: n in [1..50]]; // _Wesley Ivan Hurt_, May 13 2014
%o (PARI) a(n) = 4*n^2 - 4*n + 15 \\ _Charles R Greathouse IV_, May 14 2014
%Y Cf. A000012, A059100, A114949, A222465, A259555.
%K nonn,easy
%O 1,1
%A _Aaron David Fairbanks_, May 13 2014
%E More terms from _Wesley Ivan Hurt_, May 13 2014
%E More terms and links from _Robert G. Wilson v_, May 13 2014
%E Edited: Name reformulated (with consent of the author). - _Wolfdieter Lang_, Jul 01 2015
%E Edited by _N. J. A. Sloane_, Jan 02 2020, simplifying the definition and adding a reference to the fact that this sequence arose in a sangaku problem from 1788 in a temple in Tokyo Prefecture.
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