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A233584 Coefficients of the generalized continued fraction expansion sqrt(e) = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

%I A233584 #17 Mar 25 2025 15:02:34
%S A233584 1,1,1,1,5,9,17,109,260,2909,3072,3310,3678,6715,35175,37269,439792,
%T A233584 1400459,1472451,4643918,5683171,44850176,62252861,145631385,
%U A233584 154435765,371056666,1685980637,11196453405,14795372939
%N A233584 Coefficients of the generalized continued fraction expansion sqrt(e) = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).
%C A233584 For more details on Blazys' expansions, see A233582.
%C A233584 Compared with simple continued fraction expansion for sqrt(e), this sequence starts soon growing very rapidly.
%H A233584 Stanislav Sykora, <a href="/A233584/b233584.txt">Table of n, a(n) for n = 1..1000</a>
%H A233584 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 A233584 S. Sykora, <a href="http://oeis.org/wiki/File:BlazysExpansions.txt">PARI/GP scripts for Blazys expansions and fractions</a>, OEIS Wiki
%F A233584 sqrt(e) = 1+1/(1+1/(1+1/(1+1/(5+5/(9+9/(17+17/(109+...))))))).
%t A233584 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[Sqrt@E, 35] (* _Robert G. Wilson v_, May 22 2014 *)
%o A233584 (PARI) bx(x, nmax)={local(c, v, k); \\ Blazys expansion function
%o A233584 v = vector(nmax); c = x; for(k=1, nmax, v[k] = floor(c); c = v[k]/(c-v[k]); ); return (v); }
%o A233584 bx(exp(1/2), 100) \\ Execution; use high real precision
%Y A233584 Cf. A019774 (sqrt(e)), A058281 (simple continued fraction).
%Y A233584 Cf. Blazys' expansions: A233582 (Pi), A233583, A233585, A233586, A233587 and Blazys' continued fractions: A233588, A233589, A233590, A233591.
%K A233584 nonn
%O A233584 1,5
%A A233584 _Stanislav Sykora_, Jan 06 2014