A242412 a(n) = (2*n-1)^2 + 14.

15, 23, 39, 63, 95, 135, 183, 239, 303, 375, 455, 543, 639, 743, 855, 975, 1103, 1239, 1383, 1535, 1695, 1863, 2039, 2223, 2415, 2615, 2823, 3039, 3263, 3495, 3735, 3983, 4239, 4503, 4775, 5055, 5343, 5639, 5943, 6255, 6575, 6903, 7239, 7583, 7935, 8295, 8663, 9039, 9423, 9815

Offset

1,1

Comments

The previous definition was "a(n) = normalized inverse radius of the inscribed circle that is tangent to the left circle of the symmetric arbelos and the n-th and (n-1)-st circles in the Pappus chain".

See links section for image of these circles, via Wolfram MathWorld (there an asymmetric arbelos is shown).

The Rothman-Fukagawa article has another picture of the circles, based on a Japanese 1788 sangaku problem. - N. J. A. Sloane, Jan 02 2020

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..1000

Brady Haran and Simon Pampena, Epic Circles, Numberphile video (2014).

Tony Rothman and Hidetoshi Fukagawa, Japanese temple geometry, Scientific American, Vol. 278, No. 5, May 1998, pp. 85-91.

Eric Weisstein's World of Mathematics, Image of inscribed circles (in red).

Eric Weisstein's World of Mathematics, Pappus Chain.

Wikipedia, Pappus chain.

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

Formula

a(n) = 4*n^2 - 4*n + 15.

From Colin Barker, May 14 2014: (Start)

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

G.f.: -x*(15*x^2 - 22*x + 15)/(x-1)^3. (End)

From Descartes three circle theorem:

a(n) = 2 + c(n) + c(n-1) + 2*sqrt(2*(c(n) + c(n-1) + c(n)*c(n-1)), with c(n) = A059100(n) = n^2 + 2, n >= 1, which produces 4*n^2 - 4*n + 15. - Wolfdieter Lang, Jul 01 2015

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

E.g.f.: exp(x)*(4*x^2 + 15) - 15.

a(n) = A060747(n)^2 + 14. (End)

Example

For n = 1, the radius of the outermost circle divided by the radius of a circle drawn tangent to all three of the initial inner circle, the opposite inner circle (the 0th circle in the chain), and the 1st circle in the chain is 15.
For n = 2, the radius of the outermost circle divided by the radius of a circle drawn tangent to all three of the initial inner circle, the 1st circle in the chain, and the 2nd circle in the chain is 23.

Maple

A242412:=n->4*n^2 - 4*n + 15; seq(A242412(n), n=1..50); # Wesley Ivan Hurt, May 13 2014

Mathematica

Table[4 n^2 - 4 n + 15, {n, 50}] (* Wesley Ivan Hurt, May 13 2014 *)
LinearRecurrence[{3, -3, 1}, {15, 23, 39}, 50] (* Harvey P. Dale, Feb 22 2023 *)

Prog

(Magma) [4*n^2 - 4*n + 15: n in [1..50]]; // Wesley Ivan Hurt, May 13 2014
(PARI) a(n) = 4*n^2 - 4*n + 15 \\ Charles R Greathouse IV, May 14 2014

Crossrefs

Cf. A000012, A060747, A059100, A114949, A222465, A259555.

Sequence in context: A129387 A341170 A171167 * A364729 A195036 A111151

Adjacent sequences: A242409 A242410 A242411 * A242413 A242414 A242415

Keyword

nonn,easy

Author

Aaron David Fairbanks, May 13 2014

Extensions

More terms from Wesley Ivan Hurt, May 13 2014

More terms and links from Robert G. Wilson v, May 13 2014

Edited: Name reformulated (with consent of the author). - Wolfdieter Lang, Jul 01 2015

Edited by N. J. A. Sloane, Jan 02 2020, simplifying the definition and adding a reference to the fact that this sequence arose in a sangaku problem from 1788 in a temple in Tokyo Prefecture.

Status

approved