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Search: id:A006221
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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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+0
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; listen)
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OFFSET
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0,1
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
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.
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LINKS
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Index entries for two-way infinite sequences
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.
P. Flajolet, B. Vallee and I. Vardi, Continued fractions from Euclid to the present day.
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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 S. Plouffe in his 1992 dissertation.]
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PROGRAM
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(PARI) a(n)=34*n^3+51*n^2+27*n+5
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CROSSREFS
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Sequence in context: A109057 A080988 A156514 this_sequence A144998 A067359 A156962
Adjacent sequences: A006218 A006219 A006220 this_sequence 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 (njas(AT)research.att.com).
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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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