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A006221
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From Apery continued fraction for zeta(3): zeta(3)=6/(5-1^6/(117-2^6/(535-3^6/(1463...))).
(Formerly M4026)
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3
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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
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OFFSET
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0,1
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REFERENCES
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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).
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LINKS
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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.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
Index entries for two-way infinite sequences
Index entries for zeta function.
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FORMULA
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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^2+17*n+5)=-a(-1-n).
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EXAMPLE
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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...
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MAPLE
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A006221:=z*(z+1)*(5*z**2+92*z+5)/(z-1)**4; [Conjectured by Simon Plouffe in his 1992 dissertation.]
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MATHEMATICA
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a[n_] := (2n + 1)(17n^2 + 17n + 5);
a /@ Range[0, 31] (* Jean-François Alcover, Sep 03 2019 *)
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PROG
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(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
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CROSSREFS
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Cf. A005259.
Sequence in context: A156514 A319392 A268606 * A265977 A208387 A144998
Adjacent sequences: A006218 A006219 A006220 * A006222 A006223 A006224
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Typo in description corrected Apr 09 2006 (1436 should have been 1463). Thanks to Simon Plouffe for this correction.
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STATUS
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approved
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