A352741 -id:A352741 - cp's OEIS frontend

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

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

A137515 Maximal number of right triangles in n turns of Pythagoras's snail.

Original entry on oeis.org

16, 53, 109, 185, 280, 395, 531, 685, 860, 1054, 1268, 1502, 1756, 2029, 2322, 2635, 2967, 3319, 3691, 4083, 4494, 4926, 5376, 5847, 6337, 6848, 7377, 7927, 8496, 9086, 9694, 10323, 10971, 11639, 12327, 13035, 13762, 14509, 15276, 16062, 16868, 17694

Offset: 1

Sébastien Dumortier, Apr 23 2008, Apr 25 2008

Pythagoras's snail: begin the snail with an isosceles triangle (side = 1 unit). Then new triangle's right-angle sides are the previous hypotenuse and 1 unit length side.
From one term to the next one, the number added grows by 18, 19, 20 or 21 (tested up to 5000 terms).
To restate the comment immediately above: the second differences of the terms of the sequence consist of 18, 19, 20, or 21. - Harvey P. Dale, May 20 2019

			17 triangles are needed to close the first turn. So there are 16 triangles in this turn. From the beginning, there are 53 triangles before closing the second turn... etc.

Wikipedia, Spiral of Theodorus.

Cf. A072895, A352741.

w[n_] := ArcSin[ 1/Sqrt[n+1] ]//N; s[1] = w[1]; s[n_] := s[n] = s[n-1] + w[n]; a[n_] := (an = 1; While[ s[an] < 2*Pi*n, an++]; an-1); Table[ an = a[n]; Print[an]; an, {n, 1, 42}] (* Jean-François Alcover, Feb 24 2012 *)

from math import asin, sqrt, pi
hyp=2
som=0
n=1
while n<500:
    if som+asin(1/sqrt(hyp))/pi*180>n*360:
        print(hyp-2, end=', ')
        n=n+1
    som=som+asin(1/sqrt(hyp))/pi*180
    hyp=hyp+1

a(n) = A072895(n) - 1. - Robert G. Wilson v, Feb 27 2013

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula