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URL: https://oeis.org/A053518

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A053518
Numerators of successive convergents to continued fraction 1+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/(9+9/10+...))))))).
6
1, 5, 23, 45, 925, 7285, 7195, 641075, 6993545, 27779915, 1077005935, 15001154095, 6788401045, 3570274674605, 60484653310955, 40198648188145, 1869525647793155, 31559031031400605, 2865359642850975565
OFFSET
0,2
COMMENTS
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.
M. A. Stern, Theorie der Kettenbrüche und ihre Anwendung, Crelle, 1832, pp. 1-22.
LINKS
Leonhard Euler, 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.
EXAMPLE
Convergents are 1, 5/3, 23/15, 45/29, 925/597, 7285/4701, ...
MAPLE
for j from 1 to 50 do printf(`%d, `, numer(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_] := Numerator[num[n]/den[n]]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jan 16 2013 *)
CROSSREFS
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 Sellers, Feb 02 2000
STATUS
approved