A013696 -id:A013696

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A013631

Continued fraction for zeta(3).

+10
31

1, 4, 1, 18, 1, 1, 1, 4, 1, 9, 9, 2, 1, 1, 1, 2, 7, 1, 1, 7, 11, 1, 1, 1, 3, 1, 6, 1, 30, 1, 4, 1, 1, 4, 1, 3, 1, 2, 7, 1, 3, 1, 2, 2, 1, 16, 1, 1, 3, 3, 1, 2, 2, 1, 6, 1, 1, 1, 6, 1, 1, 4, 428, 5, 1, 1, 3, 1, 1, 11, 2, 4, 4, 5, 4, 1, 5, 14, 1, 3, 1, 2, 19, 1, 2, 5, 1, 7, 1, 1, 1, 1, 1, 57, 3, 2, 14, 2 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`Harry J. Smith, Table of n, a(n) for n = 0..19999` `Eric Weisstein's World of Mathematics, Apery's Constant.` `G. Xiao, Contfrac.` `Index entries for continued fractions for constants.` `Index entries for zeta function.`
	EXAMPLE	`zeta(3) = 1.2020569031595942... = 1 + 1/(4 + 1/(1 + 1/(18 + 1/(1 + ...)))). - Harry J. Smith, Apr 20 2009`
	MATHEMATICA	`ContinuedFraction[ Zeta[3], 100]`
	PROG	`(PARI) { allocatemem(932245000); default(realprecision, 21000); x=contfrac(zeta(3)); for (n=1, 20000, write("b013631.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 20 2009`
	CROSSREFS	`Cf. A002117 (decimal expansion), A078984, A078985 (convergents).` `Cf. continued fractions for zeta(2)-zeta(20): A013679, A013680-A013696.`
	KEYWORD	`nonn,cofr,nice`
	AUTHOR	`N. J. A. Sloane, John Morrison (John.Morrison(AT)armltd.co.uk)`
	EXTENSIONS	`Offset changed by Andrew Howroyd, Jul 10 2024`
	STATUS	`approved`

A013679

Continued fraction for zeta(2) = Pi^2/6.

+10
28

1, 1, 1, 1, 4, 2, 4, 7, 1, 4, 2, 3, 4, 10, 1, 2, 1, 1, 1, 15, 1, 3, 6, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 3, 1, 1, 5, 1, 2, 2, 1, 1, 6, 27, 20, 3, 97, 105, 1, 1, 1, 1, 1, 45, 2, 8, 19, 1, 4, 1, 1, 3, 1, 2, 1, 1, 1, 5, 1, 1, 2, 3, 6, 1, 1, 1, 2, 1, 5, 1, 1, 2, 9, 5, 3, 2, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,5`
	REFERENCES	`M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.` `David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997, page 23.`
	LINKS	`T. D. Noe, Table of n, a(n) for n = 0..9999` `M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].` `G. Xiao, Contfrac` `Index entries for continued fractions for constants` `Index entries for zeta function.`
	EXAMPLE	`1.644934066848226436472415166... = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(4 + ...))))`
	MATHEMATICA	`ContinuedFraction[ Pi^2/6, 100]`
	PROG	`(PARI) { allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi^2/6); for (n=1, 20000, write("b013679.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 29 2009`
	CROSSREFS	`Cf. A013661 (decimal expansion).` `Cf. continued fractions for zeta(3)-zeta(20): A013631, A013680-A013696.`
	KEYWORD	`nonn,cofr,nice,easy`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`Offset changed by Andrew Howroyd, Jul 10 2024`
	STATUS	`approved`

A013680

Continued fraction for zeta(4).

+10
23

1, 12, 6, 1, 3, 1, 4, 183, 1, 1, 2, 1, 3, 1, 1, 5, 4, 2, 7, 23, 1, 1, 1, 1, 3, 2, 4, 2, 2, 22, 1, 13, 5, 1, 4, 2, 1, 3, 1, 1, 1, 6, 11, 40, 1, 7, 5, 2, 4, 1, 2, 3, 14, 9, 1, 33, 78, 1, 12, 4, 1, 2, 551, 1, 1, 1, 1, 1, 1, 2, 1, 9, 2, 7, 3, 1, 3, 2, 15, 1, 1, 2, 2 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`T. D. Noe, Table of n, a(n) for n = 0..9999` `G. Xiao, Contfrac` `Index entries for continued fractions for constants` `Index entries for zeta function`
	EXAMPLE	`zeta(4) = 1 + 1/(12 + 1/(6 + 1/(1 + 1/(3 + ...)))). - Harry J. Smith, Apr 29 2009`
	MATHEMATICA	`ContinuedFraction[Zeta[4], 80] (* Harvey P. Dale, Oct 13 2013 *)`
	PROG	`(PARI) { allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi^4/90); for (n=1, 20000, write("b013680.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 29 2009`
	CROSSREFS	`Cf. A013662 (zeta(4)). - Harry J. Smith, Apr 29 2009` `Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013681-A013696.`
	KEYWORD	`nonn,cofr`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`Offset changed by Andrew Howroyd, Jul 09 2024`
	STATUS	`approved`

A013681

Continued fraction for zeta(5).

+10
5

1, 27, 12, 1, 1, 15, 1, 5, 1, 2, 19, 1, 1, 32, 1, 13, 1, 1, 1, 3, 1, 3, 2, 16, 1, 12, 4, 1, 5, 1, 1, 1, 1, 1, 2, 2, 6, 1, 8, 8, 6, 2, 3, 2, 2, 1, 30, 1, 17, 116, 1, 7, 1, 1, 1, 1, 1, 1, 2, 2, 12, 1, 4, 1, 1, 94, 1, 1, 3, 3, 6, 6, 1, 1, 2, 1, 17, 1, 1, 7, 1, 1, 11 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`T. D. Noe, Table of n, a(n) for n = 0..9999` `Index entries for continued fractions for constants` `Index entries for zeta function`
	MATHEMATICA	`ContinuedFraction[Zeta[5], 80] (* Harvey P. Dale, Nov 30 2012 *)`
	CROSSREFS	`Cf. A013663 (decimal expansion).` `Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.`
	KEYWORD	`nonn,cofr`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`Offset changed by Andrew Howroyd, Jul 09 2024`
	STATUS	`approved`

A013682

Continued fraction for zeta(6).

+10
2

1, 57, 1, 1, 1, 15, 1, 6, 3, 61, 1, 5, 3, 1, 6, 1, 3, 3, 6, 1, 10, 1, 3, 2, 1, 4, 1, 1, 5, 1, 61, 1, 3, 1, 2, 1, 3, 2, 1, 3, 1, 2, 2, 28, 1, 2, 18, 53, 2, 1, 17, 11, 3, 4, 3, 5, 2, 1, 27, 9, 8, 3, 3, 3, 9, 5, 1, 3, 29, 1, 4, 1, 2, 40, 4, 8, 1, 3, 1, 2, 2, 1, 4 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`T. D. Noe, Table of n, a(n) for n = 0..9999` `Index entries for continued fractions for constants` `Index entries for zeta function`
	MATHEMATICA	`ContinuedFraction[Zeta[6], 100] (* Harvey P. Dale, Jul 19 2019 *)`
	CROSSREFS	`Cf. A013664 (decimal expansion).` `Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.`
	KEYWORD	`nonn,cofr`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`Offset changed by Andrew Howroyd, Jul 09 2024`
	STATUS	`approved`

A013683

Continued fraction for zeta(7).

+10
2

1, 119, 1, 3, 2, 1, 2, 1, 39, 2, 3, 12, 3, 1, 1, 1, 2, 6, 5, 1, 5, 1, 2, 1, 23, 2, 1, 5, 34, 2, 1, 1, 3, 47, 2, 1, 8, 16, 1, 4, 1, 2, 1, 1, 1, 10, 72, 1, 1, 1, 1, 1, 2, 3, 13, 1, 2, 1, 5, 1, 27, 2, 9283, 1, 36, 1, 1, 1, 1, 3, 3, 23, 27, 5, 2, 4, 1, 3, 16, 1, 4 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`T. D. Noe, Table of n, a(n) for n = 0..9999` `Index entries for continued fractions for constants` `Index entries for zeta function`
	MATHEMATICA	`ContinuedFraction[Zeta[7], 100] (* Harvey P. Dale, Sep 13 2020 *)`
	CROSSREFS	`Cf. A013665 (decimal expansion).` `Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.`
	KEYWORD	`nonn,cofr`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`Offset changed by Andrew Howroyd, Jul 09 2024`
	STATUS	`approved`

A013684

Continued fraction for zeta(8).

+10
2

1, 245, 3, 1, 8, 4, 2, 3, 2, 1, 1, 4, 1, 3, 12, 2, 2, 34, 1, 1, 1, 1, 4, 9, 1, 56, 3, 38, 1, 1, 6, 1, 1, 1, 1, 3, 2, 1, 1, 5, 9, 3, 1, 11, 2, 3, 1, 5, 2, 2, 1, 4, 1, 27, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 72, 17, 1, 36, 1, 5, 6, 1, 4, 10, 1, 4, 1, 4, 1, 1, 1, 8 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`T. D. Noe, Table of n, a(n) for n = 0..9999` `Index entries for continued fractions for constants` `Index entries for zeta function`
	MATHEMATICA	`ContinuedFraction[Zeta[8], 80] (* Harvey P. Dale, Aug 14 2020 *)`
	CROSSREFS	`Cf. A013666 (decimal expansion).` `Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.`
	KEYWORD	`nonn,cofr`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`Offset changed by Andrew Howroyd, Jul 09 2024`
	STATUS	`approved`

A013685

Continued fraction for zeta(9).

+10
2

1, 497, 1, 10, 5, 1, 1, 8, 3, 2, 2, 1, 2, 1, 2, 5, 4, 2, 49, 1, 3, 3, 1, 1, 2, 1, 2, 30, 4, 1, 17, 3, 8, 2, 1, 2, 1, 1, 10, 6, 9, 2, 3, 1, 22, 1, 2, 1, 1, 2, 1, 1, 2, 18, 1, 1, 1, 9, 1, 2, 9, 1, 5, 2, 4, 1, 5, 1, 2, 2, 2, 6, 1, 8, 1, 5, 1, 4, 1483, 1, 3, 1, 2, 7 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`T. D. Noe, Table of n, a(n) for n = 0..9999` `Index entries for continued fractions for constants` `Index entries for zeta function`
	MATHEMATICA	`ContinuedFraction[Zeta[9], 100] (* Harvey P. Dale, Aug 05 2023 *)`
	CROSSREFS	`Cf. A013667 (decimal expansion).` `Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.`
	KEYWORD	`nonn,cofr`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`Offset changed by Andrew Howroyd, Jul 09 2024`
	STATUS	`approved`

A013695

Continued fraction for zeta(19).

+10
2

1, 524050, 1, 1, 2, 3, 1, 1, 1, 1, 3, 1, 2, 5, 14, 1, 5, 1, 3, 1, 3, 1, 3, 2, 1, 1, 1, 4, 1, 9, 1, 2, 1, 6, 2, 1, 1, 1, 76, 85, 1, 8, 1, 1, 7, 12, 7, 2, 1, 2, 4, 1, 3, 1, 22, 1, 3, 6, 1, 1, 1, 1, 1, 4, 1, 11, 1, 3, 1, 2, 1, 6, 1, 9, 1, 2 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`Paolo Xausa, Table of n, a(n) for n = 0..9999` `Index entries for continued fractions for constants` `Index entries for zeta function`
	MATHEMATICA	`ContinuedFraction[Zeta[19], 100] (* Paolo Xausa, Jul 03 2024 *)`
	CROSSREFS	`Cf. A013677 (decimal expansion).` `Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.`
	KEYWORD	`nonn,cofr`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`Offset changed by Andrew Howroyd, Aug 10 2024`
	STATUS	`approved`

A013686

Continued fraction for zeta(10).

+10
1

1, 1005, 2, 4, 1, 98, 7, 11, 2, 1, 1, 6, 2, 3, 28, 1, 37, 1, 2, 7, 9, 13, 85, 4, 3, 34, 5, 3, 7, 4, 7, 1, 3, 2, 1, 22, 1, 1, 1, 1, 3, 15, 1, 9, 12, 1, 3, 3, 3, 1, 3, 2, 1, 2, 1, 1, 2, 10, 8, 2, 2, 11, 54, 4, 5, 1, 2, 2, 1, 3, 2, 1, 19, 4, 5, 1, 2, 2, 7, 1, 200 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`Paolo Xausa, Table of n, a(n) for n = 0..9999` `Index entries for continued fractions for constants` `Index entries for zeta function`
	MATHEMATICA	`ContinuedFraction[Zeta[10], 100] (* Paolo Xausa, Jul 03 2024 *)`
	CROSSREFS	`Cf. A013668 (decimal expansion).` `Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.`
	KEYWORD	`nonn,cofr`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`Offset changed by Andrew Howroyd, Jul 09 2024`
	STATUS	`approved`

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Last modified September 1 06:23 EDT 2024. Contains 375575 sequences. (Running on oeis4.)