

A072895


Least k for the Theodorus spiral to complete n revolutions.


9



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

1,1


COMMENTS

"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 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
The preceding constant, c, is actually is K/2, where K is the Hlawka's Schneckenkonstante (A105459).  Robert G. Wilson v, Jul 10 2013


REFERENCES

P. J. Davis, Spirals from Theodorus to Chaos, A K Peters, Wellesley, MA, 1993.
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 3334.
David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, London, England, 1997, page 76.


LINKS

Robert G. Wilson v, Table of n, a(n) for n = 1..21000
Edmund Hlawka, Gleichverteilung und Quadratwurzelschnecke, Monatsh. Math., 89 (1980) 1944. [For a summary in English see the Davis reference, pp. 157167.]
Herbert Kociemba, The Spiral of Theodorus


FORMULA

a(n) = k means k is the least integer such that Sum_{i=1..k} arctan(1/sqrt(i)) > 2n*Pi.
a(n) = A137515(n) + 1.  Robert G. Wilson v, Feb 27 2013


MATHEMATICA

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 *)
K = 2.15778299665944622; f[n_] := Floor[(n*Pi  K/2)^2  1/6]; Array[f, 41] (* Robert G. Wilson v, Jul 10 2013 *)
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 *)


CROSSREFS

Cf. A002194, A105459.
Sequence in context: A146405 A228244 A158968 * A300059 A097059 A253424
Adjacent sequences: A072892 A072893 A072894 * A072896 A072897 A072898


KEYWORD

nonn


AUTHOR

Robert G. Wilson v, Jul 29 2002


STATUS

approved



