A233586 - OEIS

%I #13 May 23 2014 15:08:54

%S 1,6,12,19,63,263,856,2632,7714,9683,888970,1200867,1691244,2350415,

%T 3433770,4482812,17544235,48509602,53801529,114221223,124712727,

%U 997393454,16681741997,17954856574,105651203040

%N 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 For more details on Blazys' expansions, see A233582.

%H Stanislav Sykora, <a href="/A233586/b233586.txt">Table of n, a(n) for n = 1..670</a>

%H 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 S. Sykora, <a href="http://oeis.org/wiki/File:BlazysExpansions.txt">PARI/GP scripts for Blazys expansions and fractions</a>, OEIS Wiki

%F 2*gamma = 1+1/(6+6/(12+12/(19+19/(63+63/(263+...))))).

%t 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 (PARI) bx(x, nmax)={local(c, v, k); // Blazys expansion function

%o v = vector(nmax); c = x; for(k=1, nmax, v[k] = floor(c); c = v[k]/(c-v[k]); ); return (v); }

%o bx(2*Euler, 670) // Execution; use very high real precision

%Y Cf. A001620 (gamma).

%Y Cf. Blazys' expansions: A233582 (Pi), A233583 (e), A233584 (sqrt(e)), A233585 (1/gamma), A233587 and Blazys' continued fractions: A233588, A233589, A233590, A233591.

%K nonn

%O 1,2

%A _Stanislav Sykora_, Jan 06 2014