A006221 From Apery continued fraction for zeta(3): zeta(3)=6/(5-1^6/(117-2^6/(535-3^6/(1463...))). (Formerly M4026)

5, 117, 535, 1463, 3105, 5665, 9347, 14355, 20893, 29165, 39375, 51727, 66425, 83673, 103675, 126635, 152757, 182245, 215303, 252135, 292945, 337937, 387315, 441283, 500045, 563805, 632767, 707135, 787113, 872905, 964715, 1062747

Offset

0,1

References

G. V. Chudnovsky, Transcendental numbers, pp. 45-69 of Number Theory Carbondale 1979, Lect. Notes Math. 751 (1982).

S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 46.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

P. Flajolet, B. Vallee and I. Vardi, Continued fractions from Euclid to the present day, preprint, 2000.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Xavier Viennot, Introduction to Chapter 3 on continued fractions, 2013

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Index entries for two-way infinite sequences

Index entries for zeta function.

Formula

G.f.: (5 + 97*x + 97*x^2 + 5*x^3)/(1-x)^4.

a(n) = 34*n^3 + 51*n^2 + 27*n + 5 = (2*n + 1)*(17*n*(n+1) + 5) [Viennot, p.2].

Can be extended to negative indices by: a(n) = -a(-1-n).

Example

zeta(3) = 1.20205690315959428539973816151...,
while eight terms of the sequence gives 6/(5-1^6/(117-2^6/(535-3^6/(1463-4^6/(3105-5^6/(9347-6^6/(14355)))))))) = 1.20205690315959366144848279245...

Maple

A006221:=z*(z+1)*(5*z**2+92*z+5)/(z-1)**4; [Conjectured by Simon Plouffe in his 1992 dissertation.]

Mathematica

a[n_] := (2n + 1)(17n^2 + 17n + 5);
a /@ Range[0, 31] (* Jean-François Alcover, Sep 03 2019 *)

Prog

(PARI) a(n)=34*n^3+51*n^2+27*n+5
(Haskell)
a006221 n = (17 * n * (n + 1) + 5) * (2 * n + 1)
-- Reinhard Zumkeller, Mar 13 2014

Crossrefs

Apéry's number or Apéry's constant zeta(3) is A002117. - N. J. A. Sloane, Jul 11 2023

Cf. A005259.

Sequence in context: A156514 A319392 A268606 * A265977 A208387 A144998

Adjacent sequences: A006218 A006219 A006220 * A006222 A006223 A006224

Keyword

nonn

Author

N. J. A. Sloane.

Extensions

Typo in description corrected Apr 09 2006 (1436 should have been 1463). Thanks to Simon Plouffe for this correction.

Status

approved