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

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

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.

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

Cf. A053519, A053520, A053556, A053557.

Sequence in context: A121308 A222088 A146467 * A154625 A107011 A031387

Adjacent sequences: A053515 A053516 A053517 * A053519 A053520 A053521

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