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A233582 Coefficients of the generalized continued fraction expansion Pi = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

%I A233582 #26 May 22 2014 15:44:15
%S A233582 3,21,111,113,158,160,211,216,525,1634,1721,7063,8771,15077,26168,
%T A233582 58447,223767,254729,587278,1046086,1491449,1635223,1689171,2039096,
%U A233582 2290214,13444599,22666443,1276179737,4470200748
%N A233582 Coefficients of the generalized continued fraction expansion Pi = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).
%C A233582 Definition of "Blazys" generalized continued fraction expansion of an irrational real number x>1:
%C A233582 Set n=1,r=x; (ii) set a(n)=floor(r); (iii) set r=a(n)/(r-a(n)); (iv) increment n and iterate from point (ii).
%C A233582 For the inverse of this mapping, see A233588.
%H A233582 Stanislav Sykora, <a href="/A233582/b233582.txt">Table of n, a(n) for n = 1..1000</a>
%H A233582 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 A233582 S. Sykora, <a href="http://oeis.org/wiki/File:BlazysExpansions.txt">PARI/GP scripts for Blazys expansions and fractions</a>, OEIS Wiki
%F A233582 Pi = 3+3/(21+21/(111+111/(113+113/(158+...)))).
%t A233582 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[Pi, 33] (* _Robert G. Wilson v_, May 22 2014 *)
%o A233582 (PARI) bx(x,nmax)={local(c,v,k);
%o A233582 v = vector(nmax);c = x;for(k=1,nmax,v[k] = floor(c);c = v[k]/(c-v[k]););return (v);}
%o A233582 bx(Pi,1000) \\ Execution; use very high real precision
%Y A233582 Cf. A000796 (Pi), A233583-A233591.
%K A233582 nonn,cofr
%O A233582 1,1
%A A233582 _Stanislav Sykora_, Jan 02 2014