A134469 -id:A134469 - cp's OEIS frontend

A134470 Continued fraction expansion of -zeta(1/2)/sqrt(2*Pi).

0, 1, 1, 2, 1, 1, 8, 1, 5, 1, 1, 1, 12, 5, 1, 1, 5, 1, 12, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 2, 2, 2, 1, 11, 1, 6, 1, 3, 2, 1, 1, 1, 1, 1, 2, 6, 7, 1, 4, 2, 1, 1, 1, 13, 1, 1, 1, 2, 4, 2, 11, 1, 2, 5, 1, 8, 1, 78, 10, 1, 64, 1, 29, 1, 3, 1, 1, 1, 2, 1, 12, 1, 2, 1, 4, 1, 2, 1, 2, 32, 1, 92, 1, 14, 1, 10, 12, 2, 3, 16, 2, 1, 1, 1, 1, 8, 3, 15, 1, 2, 2, 1, 4, 4, 2, 8, 1, 1557, 3, 1, 69, 1, 5, 3, 11, 1, 1

Offset: 0

Hans J. H. Tuenter, Oct 27 2007

G. C. Greubel, Table of n, a(n) for n = 0..10000
Hans J. H. Tuenter, Overshoot in the Case of Normal Variables: Chernoff's Integral, Latta's Observation and Wijsman's Sum, Sequential Analysis, 26(4) (2007) 481-488.

Cf. A134469 (Decimal expansion), A134471 (Numerators of continued fraction convergents), A134472 (Denominators of continued fraction convergents).

Digits:=100; cfrac(-Zeta(1/2)/sqrt(2*Pi),30,'quotients');

ContinuedFraction[ -Zeta[1/2]/Sqrt[2 \[Pi]], 100] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)

default(realprecision,1000);
c=-zeta(1/2)/sqrt(2*Pi); /* == 0.582597157... (A134469) */
contfrac(c) /* gives 967 terms */

More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

A134471 Numerators of the convergents of the continued fraction expansion of -zeta(1/2)/sqrt(2*Pi).

Original entry on oeis.org

0, 1, 1, 3, 4, 7, 60, 67, 395, 462, 857, 1319, 16685, 84744, 101429, 186173, 1032294, 1218467, 15653898, 16872365, 32526263, 49398628, 81924891, 213248410, 295173301, 508421711, 803595012, 1312016723, 3427628458, 11594902097, 26617432652, 64829767401

Offset: 1

Hans J. H. Tuenter, Oct 27 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
Hans J. H. Tuenter, Overshoot in the Case of Normal Variables: Chernoff's Integral, Latta's Observation and Wijsman's Sum, Sequential Analysis, 26(4) (2007) 481-488.

Cf. A134469 (Decimal expansion), A134470 (Continued fraction expansion), A134472 (Denominators of continued fraction convergents).

Numerator[Convergents[-Zeta[1/2]/Sqrt[2Pi],30]] (* Harvey P. Dale, Sep 07 2015 *)

More terms from Harvey P. Dale, Sep 07 2015

A134472 Denominators of the convergents of the continued fraction expansion of -zeta(1/2)/sqrt(2*Pi).

Original entry on oeis.org

1, 1, 2, 5, 7, 12, 103, 115, 678, 793, 1471, 2264, 28639, 145459, 174098, 319557, 1771883, 2091440, 26869163, 28960603, 55829766, 84790369, 140620135, 366030639, 506650774, 872681413, 1379332187, 2252013600, 5883359387, 19902091761, 45687542909, 111277177579, 268241898067

Offset: 13

Hans J. H. Tuenter, Oct 27 2007

G. C. Greubel, Table of n, a(n) for n = 13..1012
Hans J. H. Tuenter, Overshoot in the Case of Normal Variables: Chernoff's Integral, Latta's Observation and Wijsman's Sum, Sequential Analysis, 26(4) (2007) 481-488.

Cf. A134469 (Decimal expansion), A134470 (Continued fraction expansion), A134471 (Numerators of continued fraction convergents).

Denominator[Convergents[-Zeta[1/2]/Sqrt[2 Pi], 50]] (* G. C. Greubel, Mar 28 2018 *)

Terms a(33) onward added by G. C. Greubel, Mar 28 2018

A096616 Decimal expansion of 2/3 + zeta(1/2)/sqrt(2*Pi).

Original entry on oeis.org

0, 8, 4, 0, 6, 9, 5, 0, 8, 7, 2, 7, 6, 5, 5, 9, 9, 6, 4, 6, 1, 4, 8, 9, 5, 0, 2, 4, 7, 9, 0, 3, 5, 5, 1, 1, 9, 3, 7, 5, 7, 2, 7, 9, 6, 4, 6, 8, 0, 1, 1, 9, 6, 1, 8, 4, 2, 9, 7, 2, 7, 2, 4, 6, 0, 0, 1, 3, 5, 9, 7, 9, 0, 7, 0, 1, 6, 7, 7, 2, 0, 6, 2, 4, 8, 7, 4, 7, 5, 9, 8, 3, 1, 8, 9, 0, 6, 3, 6, 0, 9, 8

Offset: 0

Eric W. Weisstein, Jun 30 2004

			0.0840695087...

David H. Bailey, Jonathan M. Borwein, Neil J. Calkin, Roland Girgensohn, D. Russell Luke and Victor H. Moll, Experimental Mathematics in Action, Wellesley, MA: A K Peters, 2007, pp. 18 and 227.
Jonathan Borwein, David Bailey and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, Wellesley, MA: A K Peters, 2004, pp. 15-17.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Jonathan M. Borwein and Scott B. Lindstrom, Meetings with Lambert W and other special functions in optimization and analysis, Pure and Applied Functional Analysis, Vol. 1, No. 3 (2016), pp. 361-396, alternative link.
Donald E. Knuth, Problem 10832, The American Mathematical Monthly, Vol. 107, No. 9 (2000), p. 863, A Stirling Series, solution by Cecil C. Rousseau, ibid., Vol. 108, No. 9 (2001), pp. 877-878.
Allen Stenger, Experimental Math for Math Monthly Problems, The American Mathematical Monthly, Vol. 124, No. 2 (2017), pp. 116-131, alternative link.
Eric Weisstein's World of Mathematics, Knuth's Series.

Cf. A019727, A059750, A231863.

Flatten[{0, RealDigits[2/3 + Zeta[1/2]/Sqrt[2*Pi], 10, 100][[1]]}] (* Vaclav Kotesovec, Aug 16 2015 *)

2/3 + zeta(1/2)/sqrt(2*Pi) \\ Michel Marcus, Aug 15 2015

Equals Sum_{k>=1} (1/sqrt(2*Pi*k) - k^k/(k!*exp(k))). - Amiram Eldar, Oct 13 2020

Equals 2/3 - A134469. - R. J. Mathar, Dec 17 2024

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A134470 Continued fraction expansion of -zeta(1/2)/sqrt(2*Pi).

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A134471 Numerators of the convergents of the continued fraction expansion of -zeta(1/2)/sqrt(2*Pi).

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A134472 Denominators of the convergents of the continued fraction expansion of -zeta(1/2)/sqrt(2*Pi).

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A096616 Decimal expansion of 2/3 + zeta(1/2)/sqrt(2*Pi).

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