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

%I A006221 M4026 #52 Dec 29 2024 01:06:59
%S A006221 5,117,535,1463,3105,5665,9347,14355,20893,29165,39375,51727,66425,
%T A006221 83673,103675,126635,152757,182245,215303,252135,292945,337937,387315,
%U A006221 441283,500045,563805,632767,707135,787113,872905,964715,1062747,1167205,1278293,1396215,1521175
%N A006221 From Apery continued fraction for zeta(3): zeta(3)=6/(5-1^6/(117-2^6/(535-3^6/(1463...)))).
%D A006221 G. V. Chudnovsky, Transcendental numbers, pp. 45-69 of Number Theory Carbondale 1979, Lect. Notes Math. 751 (1982).
%D A006221 S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 46.
%D A006221 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006221 Reinhard Zumkeller, <a href="/A006221/b006221.txt">Table of n, a(n) for n = 0..10000</a>
%H A006221 P. Flajolet, B. Vallee, and I. Vardi, <a href="http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/vardi3.pdf">Continued fractions from Euclid to the present day</a>, preprint, 2000.
%H A006221 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A006221 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A006221 Xavier Viennot, <a href="https://www.stat.purdue.edu/~mdw/ChapterIntroductions/ContinuedFractionsUpdateViennot.pdf">Introduction to Chapter 3 on continued fractions</a>, 2013
%H A006221 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%H A006221 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H A006221 <a href="/index/Z#zeta_function">Index entries for zeta function</a>.
%F A006221 G.f.: (5 + 97*x + 97*x^2 + 5*x^3)/(1-x)^4.
%F A006221 a(n) = 34*n^3 + 51*n^2 + 27*n + 5 = (2*n + 1)*(17*n*(n+1) + 5) [Viennot, p.2].
%F A006221 Can be extended to negative indices by: a(n) = -a(-1-n).
%F A006221 E.g.f.: exp(x)*(5 + 112*x + 153*x^2 + 34*x^3). - _Stefano Spezia_, Nov 03 2024
%e A006221 zeta(3) = 1.20205690315959428539973816151...,
%e A006221 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...
%p A006221 A006221:=z*(z+1)*(5*z**2+92*z+5)/(z-1)**4; [Conjectured by _Simon Plouffe_ in his 1992 dissertation.]
%t A006221 a[n_] := (2n + 1)(17n^2 + 17n + 5);
%t A006221 a /@ Range[0, 31] (* _Jean-François Alcover_, Sep 03 2019 *)
%o A006221 (PARI) a(n)=34*n^3+51*n^2+27*n+5
%o A006221 (Haskell)
%o A006221 a006221 n = (17 * n * (n + 1) + 5) * (2 * n + 1)
%o A006221 -- _Reinhard Zumkeller_, Mar 13 2014
%Y A006221 Apéry's number or Apéry's constant zeta(3) is A002117. - _N. J. A. Sloane_, Jul 11 2023
%Y A006221 Cf. A005259.
%K A006221 nonn,easy
%O A006221 0,1
%A A006221 _N. J. A. Sloane_
%E A006221 Typo in description corrected Apr 09 2006 (1436 should have been 1463). Thanks to Simon Plouffe for this correction.