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A233583 Coefficients of the generalized continued fraction expansion e = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

%I A233583 #36 Nov 23 2024 16:39:56
%S A233583 2,2,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,
%T A233583 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,
%U A233583 49,50,51,52,53,54,55,56
%N A233583 Coefficients of the generalized continued fraction expansion e = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).
%C A233583 For more details on Blazys' expansion, see A233582.
%C A233583 This sequence matches that of natural numbers (A000027), offset by 1, with two different starting terms.
%H A233583 Robert G. Wilson v, <a href="/A233583/b233583.txt">Table of n, a(n) for n = 1..1000</a>
%H A233583 Augustine O. Munagi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Munagi/munagi10.html">Integer Compositions and Higher-Order Conjugation</a>, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
%H A233583 Franck Ramaharo, <a href="https://arxiv.org/abs/1902.08989">A generating polynomial for the two-bridge knot with Conway's notation C(n,r)</a>, arXiv:1902.08989 [math.CO], 2019.
%H A233583 S. Sykora, <a href="http://dx.doi.org/10.3247/sl4math13.001">Blazys' Expansions and Continued Fractions</a>, Stans Library, Vol.IV, 2013, DOI 10.3247/sl4math13.001
%H A233583 S. Sykora, <a href="http://oeis.org/wiki/File:BlazysExpansions.txt">PARI/GP scripts for Blazys expansions and fractions</a>, OEIS Wiki
%F A233583 e = 2+2/(2+2/(2+2/(3+3/(4+4/(5+...))))).
%t A233583 BlazysExpansion[n_, mx_] := Block[{k = 1, x = n, lmt = mx + 1, s, lst = {}}, While[k < lmt, s = Floor[x]; x = 1/(x/s - 1); AppendTo[lst, s]; k++]; lst]; BlazysExpansion[E, 80] (* _Robert G. Wilson v_, May 22 2014 *)
%o A233583 (PARI)
%o A233583 default(realprecision,100);
%o A233583 bx(x,nmax)={local(c,v,k); \\ Blazys expansion function
%o A233583 v = vector(nmax);c = x;for(k=1,nmax,v[k] = floor(c);c = v[k]/(c-v[k]););return (v);}
%o A233583 bx(exp(1),100) \\ Execution; use high real precision
%Y A233583 Cf. A000027 (natural numbers), A001113 (number e).
%Y A233583 Cf. Blazys' expansions: A233582 (Pi), A233584, A233585, A233586, A233587 and Blazys' continued fractions: A233588, A233589, A233590, A233591.
%K A233583 nonn,easy
%O A233583 1,1
%A A233583 _Stanislav Sykora_, Jan 06 2014