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A233585 Coefficients of the generalized continued fraction expansion of the inverse of Euler constant, 1/gamma = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

%I A233585 #29 Mar 25 2025 15:02:23
%S A233585 1,1,2,2,2,2,4,12,39,71,83,484,1028,1447,9913,31542,526880,685669,
%T A233585 1396494,1534902,2295194,9521643,9643315,42421746,183962859,553915624,
%U A233585 557976754,6111180351,10671513549,61650520975,106532505646
%N A233585 Coefficients of the generalized continued fraction expansion of the inverse of Euler constant, 1/gamma = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).
%H A233585 Stanislav Sykora, <a href="/A233585/b233585.txt">Table of n, a(n) for n = 1..670</a>
%H A233585 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 A233585 S. Sykora, <a href="http://oeis.org/wiki/File:BlazysExpansions.txt">PARI/GP scripts for Blazys expansions and fractions</a>, OEIS Wiki
%F A233585 1/gamma = 1+1/(1+1/(2+2/(2+2/(2+2/(2+2/(4+4/(12+...))))))).
%t A233585 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[1/EulerGamma, 35] (* _Robert G. Wilson v_, May 22 2014 *)
%t A233585 BlazysExpansion[n_, mx_] := Reap[Nest[(1/(#/Sow[Floor[#]] - 1)) &, n, mx];][[-1, 1]]; BlazysExpansion[1/EulerGamma, 35] (* _Jan Mangaldan_, Jan 04 2017 *)
%o A233585 (PARI) bx(x, nmax)={local(c, v, k); \\ Blazys expansion function
%o A233585 v = vector(nmax); c = x; for(k=1, nmax, v[k] = floor(c); c = v[k]/(c-v[k]); ); return (v); }
%o A233585 bx(1/Euler, 670) \\ Execution; use very high real precision
%Y A233585 Cf. A233582.
%Y A233585 Cf. A001620 (gamma).
%Y A233585 Cf. Blazys's expansions: A233582 (Pi), A233583(e), A233584 (sqrt(e)), A233586 (2*gamma), A233587 and Blazys's continued fractions: A233588, A233589, A233590, A233591.
%K A233585 nonn
%O A233585 1,3
%A A233585 _Stanislav Sykora_, Jan 06 2014