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A242412
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a(n) = (2n-1)^2 + 14.
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3
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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
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1,1
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COMMENTS
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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
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LINKS
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Brady Haran and Simon Pampena, Epic Circles, Numberphile video (2014).
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {15, 23, 39}, 50] (* Harvey P. Dale, Feb 22 2023 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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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.
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STATUS
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approved
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