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A233586 Coefficients of the generalized continued fraction expansion of twice the Euler constant, 2*gamma = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

%I A233586 #17 Mar 25 2025 15:02:27
%S A233586 1,6,12,19,63,263,856,2632,7714,9683,888970,1200867,1691244,2350415,
%T A233586 3433770,4482812,17544235,48509602,53801529,114221223,124712727,
%U A233586 997393454,16681741997,17954856574,105651203040
%N A233586 Coefficients of the generalized continued fraction expansion of twice the Euler constant, 2*gamma = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).
%C A233586 For more details on Blazys' expansions, see A233582.
%H A233586 Stanislav Sykora, <a href="/A233586/b233586.txt">Table of n, a(n) for n = 1..670</a>
%H A233586 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 A233586 S. Sykora, <a href="http://oeis.org/wiki/File:BlazysExpansions.txt">PARI/GP scripts for Blazys expansions and fractions</a>, OEIS Wiki
%F A233586 2*gamma = 1+1/(6+6/(12+12/(19+19/(63+63/(263+...))))).
%t A233586 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[2 EulerGamma, 29] (* _Robert G. Wilson v_, May 22 2014 *)
%o A233586 (PARI) bx(x, nmax)={local(c, v, k); \\ Blazys expansion function
%o A233586 v = vector(nmax); c = x; for(k=1, nmax, v[k] = floor(c); c = v[k]/(c-v[k]); ); return (v); }
%o A233586 bx(2*Euler, 670) \\ Execution; use very high real precision
%Y A233586 Cf. A001620 (gamma).
%Y A233586 Cf. Blazys' expansions: A233582 (Pi), A233583 (e), A233584 (sqrt(e)), A233585 (1/gamma), A233587 and Blazys' continued fractions: A233588, A233589, A233590, A233591.
%K A233586 nonn
%O A233586 1,2
%A A233586 _Stanislav Sykora_, Jan 06 2014