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 A226317 Decimal expansion of the constant of Theodorus. 3
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The decimal expansion of the Sum {k>=1} 1/(k^(3/2) + k^(1/2)). This constant was first identified by Professor Philip J. Davis. 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). REFERENCES Julian R. Havil, The Irrationals: A Story of the Numbers You Can't Count On, Princeton University Press, Princeton NJ, 2012, page 277. LINKS Robert G. Wilson v, Table of n, a(n) for n = 1..1024 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. Steven Finch, Constant of Theodorus Walter Gautschi, Purdue University, The Spiral of Theodorus, Numerical Analysis, and Special Functions Kevin Ryde, Math-PlanePath, Theodorus Spiral Joerg Waldvogel, Analytic Continuation of the Theodorus Spiral Eric Weisstein's World of Mathematics, Theodorus's Constant FORMULA Sum_{k>=1} 1/(k^(3/2) + k^(1/2)). EXAMPLE 1.86002507922119030718069591571714332466652412152345149304919950359788... MATHEMATICA digits = 100; 2/Sqrt[Pi]*NIntegrate[(-Exp[t^2])*Log[1 - Exp[-t^2]] - 1, {t, 0, Infinity}, WorkingPrecision -> digits] // RealDigits[#, 10, digits]& // First (* or *) 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]] PROG (PARI) sumpos(k=1, 1/sqrt(k)/(1+k)) \\ Charles R Greathouse IV, Aug 29 2013 (PARI) sumalt(k=0, zeta(k+3/2)*(-1)^k) \\ Charles R Greathouse IV, Aug 29 2013 CROSSREFS Cf. A072895, A105459, A224269. Sequence in context: A031365 A004013 A010118 * A100121 A010526 A199473 Adjacent sequences:  A226314 A226315 A226316 * A226318 A226319 A226320 KEYWORD nonn,cons AUTHOR Walter Gautschi (wxg(AT)cs.purdue.edu), Robert G. Wilson v, and Jean-François Alcover, Apr 15 2013 STATUS approved

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