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

%I A053518 #24 Jul 02 2025 16:01:59
%S A053518 1,5,23,45,925,7285,7195,641075,6993545,27779915,1077005935,
%T A053518 15001154095,6788401045,3570274674605,60484653310955,40198648188145,
%U A053518 1869525647793155,31559031031400605,2865359642850975565
%N 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+...))))))).
%C A053518 A053518/A053519 -> (2*e-5)/(3-e) = 1.5496467783... as n-> infinity.
%D A053518 L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 562.
%D A053518 E. Maor, e: The Story of a Number, Princeton Univ. Press 1994, pp. 151 and 157.
%D A053518 M. A. Stern, Theorie der Kettenbrüche und ihre Anwendung, Crelle, 1832, pp. 1-22.
%H A053518 Leonhardo Eulero, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k69587">Introductio in analysin infinitorum. Tomus primus</a>, Lausanne, 1748.
%H A053518 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.
%e A053518 Convergents are 1, 5/3, 23/15, 45/29, 925/597, 7285/4701, ...
%p A053518 for j from 1 to 50 do printf(`%d,`,numer(cfrac([1,seq([i,i+1],i=2..j)]))); od:
%t A053518 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 *)
%Y A053518 Cf. A053519, A053520, A053556, A053557.
%K A053518 nonn,frac,nice,easy
%O A053518 0,2
%A A053518 _N. J. A. Sloane_, Jan 15 2000
%E A053518 Thanks to _R. K. Guy_, _Steven Finch_, _Bill Gosper_ for comments
%E A053518 More terms from _James Sellers_, Feb 02 2000