A259555 a(n) = 2*n^2 - 2*n + 17.

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

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