login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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)/....))). 12

%I

%S 3,21,111,113,158,160,211,216,525,1634,1721,7063,8771,15077,26168,

%T 58447,223767,254729,587278,1046086,1491449,1635223,1689171,2039096,

%U 2290214,13444599,22666443,1276179737,4470200748

%N 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 Definition of "Blazys" generalized continued fraction expansion of an irrational real number x>1:

%C 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 For the inverse of this mapping, see A233588.

%H Stanislav Sykora, <a href="/A233582/b233582.txt">Table of n, a(n) for n = 1..1000</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 Pi = 3+3/(21+21/(111+111/(113+113/(158+...)))).

%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[Pi, 33] (* _Robert G. Wilson v_, May 22 2014 *)

%o (PARI) bx(x,nmax)={local(c,v,k);

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

%o bx(Pi,1000) \\ Execution; use very high real precision

%Y Cf. A000796 (Pi), A233583-A233591.

%K nonn,cofr

%O 1,1

%A _Stanislav Sykora_, Jan 02 2014

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 11 06:25 EDT 2020. Contains 336422 sequences. (Running on oeis4.)