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A242412 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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

a(n) can also be given by the quadratic 4*n^2 - 4*n + 15.

LINKS

Table of n, a(n) for n = 1..50

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

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.

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. - Colin Barker, May 14 2014

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

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 *)

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, A059100, A114949, A222465, A259555.

Sequence in context: A014312 A129387 A171167 * A195036 A111151 A166657

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

STATUS

approved

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Last modified February 20 23:24 EST 2019. Contains 320362 sequences. (Running on oeis4.)