A053519 - OEIS

%I #26 Mar 12 2017 12:25:57

%S 1,3,15,29,597,4701,4643,413691,4512993,17926611,695000919,9680369943,

%T 4380611853,2303928046437,39031251610227,25940523189513,

%U 1206420504316107,20365306128628437,1849040492948486661

%N Denominators of successive convergents to continued fraction 1+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/(9+9/10+...))))))).

%C Also numerators of successive convergents to continued fraction 1/(2+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/9+...))))))).

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

%D L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 562.

%D E. Maor, e: The Story of a Number, Princeton Univ. Press 1994, pp. 151 and 157.

%H Leonhardo Eulero, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k69587">Introductio in analysin infinitorum. Tomus primus</a>, Lausanne, 1748.

%H L. Euler, Introduction à l'analyse infinitésimale, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k3884z">Tome premier</a>, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k38858">Tome second</a>, trad. du latin en français par J. B. Labey, Paris, 1796-1797.

%H M. A. Stern, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002138999">Theorie der Kettenbrüche und ihre Anwendung</a>, Crelle, 1832, pp. 1-22.

%e Convergents (to the first continued fraction) are 1, 5/3, 23/15, 45/29, 925/597, 7285/4701, ...

%p for j from 1 to 50 do printf(`%d,`,denom(cfrac([1,seq([i,i+1],i=2..j)]))); od:

%t 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 *)

%Y Cf. A053518, A053520, A053556, A053557.

%K nonn,frac,nice,easy

%O 0,2

%A _N. J. A. Sloane_, Jan 15 2000

%E Thanks to _R. K. Guy_, _Steven Finch_, _Bill Gosper_ for comments

%E More terms from _James A. Sellers_, Feb 02 2000