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%I #71 Dec 08 2017 03:53:47
%S 17,54,110,186,281,396,532,686,861,1055,1269,1503,1757,2030,2323,2636,
%T 2968,3320,3692,4084,4495,4927,5377,5848,6338,6849,7378,7928,8497,
%U 9087,9695,10324,10972,11640,12328,13036,13763,14510,15277,16063,16869
%N Least k for the Theodorus spiral to complete n revolutions.
%C "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.
%C 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
%C The preceding constant and function b(n) = a(n) for all n < 21001. - _Robert G. Wilson v_, Mar 07 2013; Update: b(n) = a(n) for all n < 10^9. - _Herbert Kociemba_, Jul 15 2013
%C The preceding constant, c, is actually is -K/2, where K is the Hlawka's Schneckenkonstante (A105459). - _Robert G. Wilson v_, Jul 10 2013
%D P. J. Davis, Spirals from Theodorus to Chaos, A K Peters, Wellesley, MA, 1993.
%D Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, page 35.
%D Paul J. Nahin, "An Imaginary Tale, The Story of [Sqrt(-1)]," Princeton University Press, Princeton, NJ. 1998, pgs 33-34.
%D David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, London, England, 1997, page 76.
%H Robert G. Wilson v, <a href="/A072895/b072895.txt">Table of n, a(n) for n = 1..21000</a>
%H Edmund Hlawka, <a href="http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002481898">Gleichverteilung und Quadratwurzelschnecke</a>, Monatsh. Math., 89 (1980) 19-44. [For a summary in English see the Davis reference, pp. 157-167.]
%H Herbert Kociemba, <a href="http://kociemba.org/themen/spirale/theodorus.html">The Spiral of Theodorus</a>
%F a(n) = k means k is the least integer such that Sum_{i=1..k} arctan(1/sqrt(i)) > 2n*Pi.
%F a(n) = A137515(n) + 1. - _Robert G. Wilson v_, Feb 27 2013
%t 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 *)
%t K = -2.15778299665944622; f[n_] := Floor[(n*Pi - K/2)^2 - 1/6]; Array[f, 41] (* _Robert G. Wilson v_, Jul 10 2013 *)
%t K = -2.1577829966594462209291427868295777235; a[n_] := Module[{a = -(K/2) + n Pi, b}, b = a^2 - 1/6; If[Floor[b] == Floor[b + 1/(144 a^2)], Floor[b], Undefined]] (* defined at least for all n < 10^9, _Herbert Kociemba_, Jul 15 2013 *)
%Y Cf. A002194, A105459.
%K nonn
%O 1,1
%A _Robert G. Wilson v_, Jul 29 2002
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