%I #74 Apr 01 2024 02:51:17

%S 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,

%T 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,

%U 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

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

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

%H Robert G. Wilson v, <a href="/A105459/b105459.txt">Table of n, a(n) for n = 1..16384</a>

%H David Brink, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.119.09.779">The spiral of Theodorus and sums of zeta-values at the half-integers</a>, The American Mathematical Monthly, Vol. 119, No. 9 (November 2012), pp. 779-786.

%H Edmund Hlawka, <a href="http://dx.doi.org/10.1007/BF01571563">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 Sum_{x=1..n-1} arctan(1/sqrt(x)) = 2*sqrt(n) + K + o(1). [Corrected by _M. F. Hasler_, Mar 31 2022]

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

%e -2.157782996659446220929142786829577723504139598607562455...

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

%t 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 *)

%o (PARI) sumalt(k=0,(-1)^k*zeta(k+1/2)/(2*k+1)) \\ _M. F. Hasler_, Mar 31 2022

%Y Cf. A185051 for continued fraction expansion.

%Y Cf. A072895, A137515, A352741.

%K nonn,cons

%O 1,1

%A _David Brink_, Jun 13 2011