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%I #49 Jul 19 2022 05:47:36
%S 1,8,6,0,0,2,5,0,7,9,2,2,1,1,9,0,3,0,7,1,8,0,6,9,5,9,1,5,7,1,7,1,4,3,
%T 3,2,4,6,6,6,5,2,4,1,2,1,5,2,3,4,5,1,4,9,3,0,4,9,1,9,9,5,0,3,5,9,8,3,
%U 4,2,7,2,3,3,9,9,9,2,1,3,2,0,5,6,8,8,3,8,7,5,6,4,9,9,6,1,4,4,9,5
%N Decimal expansion of the constant of Theodorus.
%C The decimal expansion of the Sum {k>=1} 1/(k^(3/2) + k^(1/2)).
%C This constant was first identified by Professor Philip J. Davis.
%C This constant is not in Steven R. Finch, Mathematical Constants, Cambridge, 2003, nor is it in the Inverse Symbolic Calculator (originally by Simon Plouffe & the Borwein brothers).
%D Philip J. Davis, Spirals: From Theodorus to Chaos, AK Peters, 1993.
%D Julian R. Havil, The Irrationals: A Story of the Numbers You Can't Count On, Princeton University Press, Princeton NJ, 2012, page 277.
%H Robert G. Wilson v, <a href="/A226317/b226317.txt">Table of n, a(n) for n = 1..1024</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 Ewan Brinkman, Robert Corless, and Veselin Jungic, <a href="https://doi.org/10.5206/mt.v1i2.14500">The Theodorus Variation</a>, Maple Transactions, Vol. 1, No. 2 (2021), Article 14500.
%H Steven Finch, <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.440.3922&rep=rep1&type=pdf">Constant of Theodorus</a>
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 663.
%H Walter Gautschi, Purdue University, <a href="http://www.cs.purdue.edu/homes/wxg/slidesTheodorus.pdf"> The Spiral of Theodorus, Numerical Analysis, and Special Functions</a>.
%H Kevin Ryde, Math-PlanePath, <a href="http://cpansearch.perl.org/src/KRYDE/Math-PlanePath-100/devel/theodorus.pl">Theodorus Spiral</a>.
%H Jörg Waldvogel, <a href="http://www.sam.math.ethz.ch/~joergw/Papers/theopaper.pdf">Analytic Continuation of the Theodorus Spiral</a>, Seminar für Angewandte Mathematik, ETH Zürich, 2008.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TheodorussConstant.html">Theodorus's Constant</a>.
%F Sum_{k>=1} 1/(k^(3/2) + k^(1/2)).
%F Equals -(2/sqrt(Pi)) * Integral_{x>=0} (exp(x^2)*log(1-exp(-x^2))+1) dx (Waldvogel, 2008). - _Amiram Eldar_, Jul 19 2022
%e 1.86002507922119030718069591571714332466652412152345149304919950359788...
%p Digits := 102: evalf(sum((k^(3/2) + k^(1/2))^(-1), k=1..infinity));
%p # _Peter Luschny_, Feb 28 2022
%t digits = 100; 2/Sqrt[Pi]*NIntegrate[(-Exp[t^2])*Log[1 - Exp[-t^2]] - 1, {t, 0, Infinity}, WorkingPrecision -> digits] // RealDigits[#, 10, digits]& // First
%t (* or *)
%t a = NSum[1/(k^(3/2) + k^(1/2)), {k, 1, Infinity}, AccuracyGoal -> 2^8, PrecisionGoal -> 2^8, WorkingPrecision -> 2^8, NSumTerms -> 2^15]; RealDigits[a, 10, 105][[1]]
%o (PARI) sumpos(k=1,1/sqrt(k)/(1+k)) \\ _Charles R Greathouse IV_, Aug 29 2013
%o (PARI) sumalt(k=0,zeta(k+3/2)*(-1)^k) \\ _Charles R Greathouse IV_, Aug 29 2013
%Y Cf. A072895, A105459, A224269.
%K nonn,cons
%O 1,2
%A Walter Gautschi (wxg(AT)cs.purdue.edu), _Robert G. Wilson v_, and _Jean-François Alcover_, Apr 15 2013
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