A072895 Least k for the Theodorus spiral to complete n revolutions.

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

Offset: 1

Author

Robert G. Wilson v, Jul 29 2002

Keywords

• nonn

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 33-34.
• 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) 19-44. [For a summary in English see the Davis reference, pp. 157-167.]
• Herbert Kociemba, The Spiral of Theodorus

Crossrefs

Cf. A002194, A105459.

Programs

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 *)

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