

A242412


a(n) = normalized inverse radius of the inscribed circle that is tangent to the left circle of the symmetric arbelos and the nth and (n1)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
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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(n1)3*a(n2)+a(n3). G.f.: x*(15*x^222*x+15) / (x1)^3.  Colin Barker, May 14 2014
From Descartes three circle theorem:
a(n) = 2 +c(n) + c(n1) + 2*sqrt(2*(c(n) + c(n1) + c(n)*c(n1)), 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



