A137515 -id:A137515 - cp's OEIS frontend

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

Robert G. Wilson v, Jul 29 2002

nonn

"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

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.

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

Cf. A002194, A105459.

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 *)

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

A105459 Decimal expansion of Hlawka's Schneckenkonstante K = -2.157782... (negated).

Original entry on oeis.org

2, 1, 5, 7, 7, 8, 2, 9, 9, 6, 6, 5, 9, 4, 4, 6, 2, 2, 0, 9, 2, 9, 1, 4, 2, 7, 8, 6, 8, 2, 9, 5, 7, 7, 7, 2, 3, 5, 0, 4, 1, 3, 9, 5, 9, 8, 6, 0, 7, 5, 6, 2, 4, 5, 5, 1, 5, 4, 8, 9, 5, 5, 5, 0, 8, 5, 8, 8, 6, 9, 6, 4, 6, 7, 9, 6, 6, 0, 6, 4, 8, 1, 4, 9, 6, 6, 9, 4, 2, 9, 8, 9, 4, 6, 3, 9, 6, 0, 8, 9, 8

Offset: 1

David Brink, Jun 13 2011

			-2.157782996659446220929142786829577723504139598607562455...

P. J. Davis, Spirals from Theodorus to Chaos, A K Peters, Wellesley, MA, 1993.

Robert G. Wilson v, Table of n, a(n) for n = 1..16384
David Brink, The spiral of Theodorus and sums of zeta-values at the half-integers, The American Mathematical Monthly, Vol. 119, No. 9 (November 2012), pp. 779-786.
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.

Cf. A185051 for continued fraction expansion.

Cf. A072895, A137515, A352741.

evalf(Sum((-1)^k*Zeta(k + 1/2)/(2*k+1), k=0..infinity), 120); # Vaclav Kotesovec, Mar 01 2016

RealDigits[ NSum[(-1)^k*Zeta[k + 1/2]/(2 k + 1), {k, 0, Infinity}, Method -> "AlternatingSigns", AccuracyGoal -> 2^6, PrecisionGoal -> 2^6, WorkingPrecision -> 2^7], 10, 2^7][[1]] (* Robert G. Wilson v, Jul 11 2013 *)

sumalt(k=0,(-1)^k*zeta(k+1/2)/(2*k+1)) \\ M. F. Hasler, Mar 31 2022

Sum_{x=1..n-1} arctan(1/sqrt(x)) = 2*sqrt(n) + K + o(1). [Corrected by M. F. Hasler, Mar 31 2022]

Equals Sum_{k>=0} (-1)^k*zeta(k+1/2)/(2*k+1). - Robert B Fowler, Oct 23 2022

A352741 Numbers on the Spiral of Theodorus that come closer to the x-axis than their neighbors; a(0) = 0 and a(1) = 1.

Original entry on oeis.org

0, 1, 7, 18, 33, 54, 80, 110, 146, 186, 231, 282, 337, 397, 462, 532, 607, 687, 772, 861, 956, 1056, 1160, 1270, 1384, 1504, 1628, 1757, 1891, 2030, 2174, 2323, 2477, 2636, 2800, 2969, 3142, 3321, 3504, 3693, 3886, 4084, 4288, 4496, 4709, 4927, 5150, 5378, 5611, 5849, 6091, 6339

Offset: 0

M. F. Hasler, Mar 31 2022

nonn

The Spiral of Theodorus (a.k.a. Pythagorean spiral or Pythagoras's snail) results from constructing the square roots sqrt(2), sqrt(3), sqrt(4), ... as hypotenuse of the right triangle having the previous one as longer leg, and the shorter leg equal to 1, starting with the segment [0, 1] for sqrt(1). In the complex plane, this construction corresponds to a sequence z(n+1) = z(n) + i*z(n)/|z(n)|, starting with z(1) = 1 (and z(0) = 0 by convention).
This sequence lists those n for which the endpoint is closer to the x-axis than the preceding and next one (plus the two initial points 0 and 1 which are the only ones lying directly on the x-axis). In terms of the complex sequence z(n), this means the indices n such that abs(Im(z(n)) <=  abs(Im(z(n +- 1))).
The two initial terms correspond to the points (0, 0) and (1, 0), which can be regarded as the start of the Pythagorean spiral, and are the only points exactly on the x-axis.
There is a smooth complex-valued function z: [0,oo) -> C, t |-> z(t), which interpolates the spiral in non-integer values. In terms of this function, a(n) = round(t(n)) where t(n) is the n-th zero of the imaginary part of z (if indexing starts with 0 for t(0) = 0, then t(1) = 1, and t(2) ~ 6.8 where z(6.8) = -1). (This function has a natural extension to the whole of R including also the negative real line, but we don't consider negative arguments here.)

Christian Mercat, Escargot de Pythagore: Spirale des racines carrées d'entiers, GeoGebra.org, Dec. 12, 2015
Wikipedia, Spiral of Theodorus, as of March 14, 2022

Cf. A072895, A137515, A105459 (Hlawka's "snail constant").

z=!t=y=0; [n | n<-[0..10^4], ![t < t=y, y > y=abs(imag(z))]*z*=1+I/abs(z)]

a(2n+1) = A072895(n) or A072895(n)+1, for all n > 0.

A172164 Differences between numbers of triangles entirely contained in two consecutive turns of Pythagoras's snail (Theodorus spiral).

Original entry on oeis.org

20, 19, 20, 19, 20, 21, 18, 21, 19, 20, 20, 20, 19, 20, 20, 19, 20, 20, 20, 19, 21, 18, 21, 19, 21, 18, 21, 19, 21, 18, 21, 19, 20, 20, 20, 19, 20, 20, 19, 20, 20, 20, 19, 20, 20, 20, 19, 21, 18, 21, 19, 20, 20, 20, 19, 20, 20, 20, 19, 20, 20, 19, 20, 20, 20, 19, 21, 18, 21

Offset: 2

Sébastien Dumortier, Jan 27 2010

nonn

Conjecture : The terms are only 18,19,20,21 (From the first thousand turns, there are 2,3% of 18, 36,5% of 19, 46,2% of 20 and 15% of 21). No period found. Probably due to Pi transcendence.
From the first one hundred thousand turns, there are 1.662% 18s, 36.350% 19s, 48.393% 20s and 13.595% 21s. - Robert G. Wilson v, Mar 31 2013
From the first 10 Million turns, there are 1.69208% 18s, 36.33984% 19s, 48.32320% 20s and 13.64488% 21s. - Herbert Kociemba, Jul 15 2013

			In the first turn, 16 triangles are complete. In the 2nd turn, there are 36 triangles completely included. The difference is 20.

Cf. A072895, A137515.

(* Obtain the sequence of A072895 and set it equal to lst. *); Differences[lst, 2] (* Robert G. Wilson v, Mar 31 2013 *)

# See A137515 for Python code, and then OooCalc for more.

The second forward difference of A072895. - Robert G. Wilson v, Mar 31 2013

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

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A105459 Decimal expansion of Hlawka's Schneckenkonstante K = -2.157782... (negated).

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A352741 Numbers on the Spiral of Theodorus that come closer to the x-axis than their neighbors; a(0) = 0 and a(1) = 1.

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A172164 Differences between numbers of triangles entirely contained in two consecutive turns of Pythagoras's snail (Theodorus spiral).

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Author

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Programs

Mathematica

Python

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