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Search: id:A053519
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| A053519 |
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Denominators of successive convergents to continued fraction 1+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/(9+9/10+...))))))). |
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+0
3
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| 1, 3, 15, 29, 597, 4701, 4643, 413691, 4512993, 17926611, 695000919, 9680369943, 4380611853, 2303928046437, 39031251610227, 25940523189513, 1206420504316107, 20365306128628437, 1849040492948486661
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Also numerators of successive convergents to continued fraction 1/(2+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/9+...))))))).
A053518/A053519 -> (2*e-5)/(3-e) = 1.5496467783... as n-> infinity.
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REFERENCES
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L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 562.
E. Maor, e: The Story of a Number, Princeton Univ. Press 1994, pp. 151 and 157.
M. A. Stern, Theorie der Kettenbr"uche und ihre Anwendung, Crelle, 1832, pp. 1-22.
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LINKS
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Leonhardo Eulero, Introductio in analysin infinitorum. Tomus primus, Lausanne, 1748.
L. Euler, Introduction a l'analyse infinitesimal Tome premier, Tome second, trad. du latin en francais par J. B. Labey, Paris, 1796-1797.
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EXAMPLE
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Convergents (to the first continued fraction) are 1, 5/3, 23/15, 45/29, 925/597, 7285/4701, ...
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MAPLE
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for j from 1 to 50 do printf(`%d, `, denom(cfrac([1, seq([i, i+1], i=2..j)]))); od:
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CROSSREFS
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Cf. A053518, A053520, A053556, A053557.
Sequence in context: A067144 A018784 A147344 this_sequence A039666 A020493 A087183
Adjacent sequences: A053516 A053517 A053518 this_sequence A053520 A053521 A053522
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KEYWORD
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nonn,frac,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Jan 15 2000
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EXTENSIONS
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Thanks to R. K. Guy, S. R. Finch, R. W. Gosper for comments.
More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 02 2000
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