

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
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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 Zetavalues at the Halfintegers, The American Mathematical Monthly, Vol. 119, No. 9 (November 2012), pp. 779786.
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, MathPlanePath, 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 JeanFrançois Alcover, Apr 15 2013


STATUS

approved



