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

Author

Kival Ngaokrajang, Jun 30 2015

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A114949, A242412 (for r0 = 1/2 = r1).

Programs

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 *)

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

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

Extensions

Edited by Wolfdieter Lang, Jun 30 2015