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)/....))).

3, 21, 111, 113, 158, 160, 211, 216, 525, 1634, 1721, 7063, 8771, 15077, 26168, 58447, 223767, 254729, 587278, 1046086, 1491449, 1635223, 1689171, 2039096, 2290214, 13444599, 22666443, 1276179737, 4470200748

Offset: 1

Author

Stanislav Sykora, Jan 02 2014

Keywords

• nonn
• cofr

Comments

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

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).

For the inverse of this mapping, see A233588.

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..1000
• S. Sykora, Blazys' Expansions and Continued Fractions, Stans Library, Vol.IV, 2013, DOI 10.3247/sl4math13.001
• S. Sykora, PARI/GP scripts for Blazys expansions and fractions, OEIS Wiki

Crossrefs

Cf. A000796 (Pi), A233583-A233591.

Programs

Mathematica

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 *)

PARI

bx(x,nmax)={local(c,v,k);
v = vector(nmax);c = x;for(k=1,nmax,v[k] = floor(c);c = v[k]/(c-v[k]););return (v);}
bx(Pi,1000) \\ Execution; use very high real precision

Formula

Pi = 3+3/(21+21/(111+111/(113+113/(158+...)))).