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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…Inf.} 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

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.

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

Steven Finch, Constant of Theodorus

Walter Gautschi, Purdue University, The Spiral of Theodorus, Numerical Analysis, and Special Functions

Kevin Ryde, Math-PlanePath, TheodorusSpiral.

Joerg Waldvogel, Analytic Continuation of the Theodorus Spiral

FORMULA

Sum_{k=1...inf} 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, A224269, A105459.

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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Last modified December 17 21:13 EST 2014. Contains 252040 sequences.