A259054 - OEIS

A259054

a(n) = 4*n^2 - 4*n + 19, n >= 1.

19, 27, 43, 67, 99, 139, 187, 243, 307, 379, 459, 547, 643, 747, 859, 979, 1107, 1243, 1387, 1539, 1699, 1867, 2043, 2227, 2419, 2619, 2827, 3043, 3267, 3499, 3739, 3987, 4243, 4507, 4779, 5059, 5347, 5643, 5947, 6259, 6579, 6907, 7243, 7587, 7939, 8299, 8667, 9043, 9427, 9819

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(n) gives twice the inverse radius of the circles touching the large Arbelos (2/3,1/3) circle (radius 1) and the n-th and (n-1)-th circles of the counterclockwise Pappus chain.

For twice the curvatures (inverse radii) of the counterclockwise Pappus chain of the (2/3,1/3) arbelos see A114949, also for the MathWorld link to Pappus chain.

For the small curvatures touching the left circle of the (2/3,1/3) arbelos and the n-th and (n-1)-st circles of the counterclockwise Pappus chain see A259555.

The curvatures of the circles can be computed with Descartes' three (actually 5) circle theorem. See A259555 for links to Descartes' theorem.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Kival Ngaokrajang, Illustration of initial terms.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = 4*n^2 - 4*n + 19, n >= 1.

O.g.f.: x*(19 - 30*x + 19*x^2)/(1 - x)^3.

From Elmo R. Oliveira, Nov 17 2024: (Start)

E.g.f.: exp(x)*(4*x^2 + 19) - 19.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

MATHEMATICA