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A072895
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Least k for the Theodorus spiral to complete n revolutions.
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4
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17, 54, 110, 186, 281, 396, 532, 686, 861, 1055, 1269, 1503, 1757, 2030, 2323, 2636, 2968, 3320, 3692, 4084, 4495, 4927, 5377, 5848, 6338, 6849, 7378, 7928, 8497, 9087, 9695, 10324, 10972, 11640, 12328, 13036, 13763, 14510, 15277, 16063, 16869
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OFFSET
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1,1
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COMMENTS
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"For n = 16 (which gives sqrt(17)) this sum is 351.15 degree, while for n = 17 the sum is 364.78 degree. That is, perhaps Theodorus stopped at sqrt(17) simply because for n > 16 his spiral started to overlap itself and the drawing became 'messy.'" Nahin p. 34.
There exists a constant c = 1.07889149832972311... such that b(n) = IntegerPart[(Pi*n + c)^2 - 1/6] differs at most by 1 from a(n) for all n>=1. At least for n<=4000 we indeed have a(n)=b(n). - Herbert Kociemba, Sep 12 2005
The preceeding constant and function b(n) = a(n) for all n < 21001. - Robert G. Wilson v, Mar 07 2013
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REFERENCES
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Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, page 35.
Paul J. Nahin, "An Imaginary Tale, The Story of [Sqrt(-1)]," Princeton University Press, Princeton, NJ. 1998, pgs 33-34.
David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, London, England, 1997, page 76.
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LINKS
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Robert G. Wilson v, Table of n, a(n) for n = 1..21000
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FORMULA
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Sum ( atan( 1/sqrt(i)), i=1..k) > 2n*Pi
a(n) = A137515(n) + 1. [Robert G. Wilson v, Feb 27 2013]
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MATHEMATICA
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s = 0; k = 1; lst = {}; Do[ While[s < (2Pi)n, (* change the value in the parentheses to change the angle *) s = N[s + ArcTan[1/Sqrt@k], 32]; k++]; AppendTo[lst, k - 1], {n, 50}]; lst (* Robert G. Wilson v, Oct 14 2012 *)
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CROSSREFS
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Cf. A002194.
Sequence in context: A146397 A146405 A158968 * A097059 A117390 A141841
Adjacent sequences: A072892 A072893 A072894 * A072896 A072897 A072898
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KEYWORD
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nonn
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AUTHOR
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Robert G. Wilson v, Jul 29 2002
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STATUS
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approved
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