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 A053519 Denominators of successive convergents to continued fraction 1+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/(9+9/10+...))))))). 4
 1, 3, 15, 29, 597, 4701, 4643, 413691, 4512993, 17926611, 695000919, 9680369943, 4380611853, 2303928046437, 39031251610227, 25940523189513, 1206420504316107, 20365306128628437, 1849040492948486661 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. REFERENCES 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. LINKS Table of n, a(n) for n=0..18. Leonhardo Eulero, Introductio in analysin infinitorum. Tomus primus, Lausanne, 1748. L. Euler, Introduction à l'analyse infinitésimale, Tome premier, Tome second, trad. du latin en français par J. B. Labey, Paris, 1796-1797. M. A. Stern, Theorie der Kettenbrüche und ihre Anwendung, Crelle, 1832, pp. 1-22. EXAMPLE Convergents (to the first continued fraction) are 1, 5/3, 23/15, 45/29, 925/597, 7285/4701, ... MAPLE for j from 1 to 50 do printf(`%d, `, denom(cfrac([1, seq([i, i+1], i=2..j)]))); od: MATHEMATICA num[0]=1; num[1]=5; num[n_] := num[n] = (n+2)*num[n-1] + (n+1)*num[n-2]; den[0]=1; den[1]=3; den[n_] := den[n] = (n+2)*den[n-1] + (n+1)*den[n-2]; a[n_] := Denominator[num[n]/den[n]]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jan 16 2013 *) CROSSREFS Cf. A053518, A053520, A053556, A053557. Sequence in context: A201434 A308385 A202506 * A039666 A020493 A087183 Adjacent sequences: A053516 A053517 A053518 * A053520 A053521 A053522 KEYWORD nonn,frac,nice,easy AUTHOR N. J. A. Sloane, Jan 15 2000 EXTENSIONS Thanks to R. K. Guy, Steven Finch, Bill Gosper for comments More terms from James A. Sellers, Feb 02 2000 STATUS approved

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Last modified December 9 04:08 EST 2023. Contains 367681 sequences. (Running on oeis4.)