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

Author

N. J. A. Sloane, Jan 15 2000

Keywords

• nonn
• frac
• nice
• easy

Comments

A053518/A053519 -> (2*e-5)/(3-e) = 1.5496467783... as n-> infinity.

Examples

			Convergents are 1, 5/3, 23/15, 45/29, 925/597, 7285/4701, ...
		

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

• 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.

Crossrefs

Cf. A053519, A053520, A053556, A053557.

Programs

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 *)

Extensions

Thanks to R. K. Guy, Steven Finch, Bill Gosper for comments

More terms from James Sellers, Feb 02 2000