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).
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).
E.g.f.: exp(x)*(5 + 112*x + 153*x^2 + 34*x^3). - Stefano Spezia, Nov 03 2024
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Typo in description corrected Apr 09 2006 (1436 should have been 1463). Thanks to Simon Plouffe for this correction.
STATUS
approved