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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
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
Eric Weisstein's World of Mathematics, Descartes Circle theorem
Eric Weisstein's World of Mathematics, Pappus chain
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: A188200 A376026 A060875 * A138600 A050845 A219396
KEYWORD
nonn,easy
AUTHOR
Kival Ngaokrajang, Jun 30 2015
EXTENSIONS
Edited by Wolfdieter Lang, Jun 30 2015
STATUS
approved