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

Author

N. J. A. Sloane, Jan 15 2000

Keywords

• nonn
• frac
• nice
• easy

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.

Examples

			Convergents (to the first continued fraction) 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.

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.
• M. A. Stern, Theorie der Kettenbrüche und ihre Anwendung, Crelle, 1832, pp. 1-22.

Crossrefs

Cf. A053518, A053520, A053556, A053557.

Programs

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 *)

Extensions

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

More terms from James Sellers, Feb 02 2000