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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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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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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).
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LINKS
| P. Flajolet, B. Vallee and I. Vardi, Continued fractions from Euclid to the present day.
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for two-way infinite sequences
Index entries for zeta function.
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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^2+17*n+5)=-a(-1-n).
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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...
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MAPLE
| A006221:=z*(z+1)*(5*z**2+92*z+5)/(z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation.]
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PROG
| (PARI) a(n)=34*n^3+51*n^2+27*n+5
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CROSSREFS
| Sequence in context: A109057 A080988 A156514 * A144998 A067359 A156962
Adjacent sequences: A006218 A006219 A006220 * A006222 A006223 A006224
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Typo in description corrected Apr 09 2006 (1436 should have been 1463). Thanks to Simon Plouffe for this correction.
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