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+...))))))).

1, 3, 15, 29, 597, 4701, 4643, 413691, 4512993, 17926611, 695000919, 9680369943, 4380611853, 2303928046437, 39031251610227, 25940523189513, 1206420504316107, 20365306128628437, 1849040492948486661

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