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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...))).
(Formerly M4026)
+0
2
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)
OFFSET

0,1

REFERENCES

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.

LINKS

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.

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).

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 S. Plouffe in his 1992 dissertation.]

PROGRAM

(PARI) a(n)=34*n^3+51*n^2+27*n+5

CROSSREFS

Sequence in context: A089638 A109057 A080988 this_sequence A144998 A067359 A065818

Adjacent sequences: A006218 A006219 A006220 this_sequence A006222 A006223 A006224

KEYWORD

nonn

AUTHOR

njas

EXTENSIONS

Typo in description corrected Apr 09 2006 (1436 should have been 1463). Thanks to Simon Plouffe for this correction.

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Last modified December 4 15:51 EST 2008. Contains 151308 sequences.


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