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
From Colin Barker, Jul 01 2015: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -x*(17*x^2 - 30*x + 17)/(x-1)^3. (End)
E.g.f.: exp(x)*(2*x^2 + 17) - 17. - Elmo R. Oliveira, Nov 17 2024
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
KEYWORD
nonn,easy
AUTHOR
Kival Ngaokrajang, Jun 30 2015
EXTENSIONS
Edited by Wolfdieter Lang, Jun 30 2015
STATUS
approved