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

1,1

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

Formula

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

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 *)

Prog

(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

Crossrefs

Cf. A000796 (Pi), A233583-A233591.

Sequence in context: A034268 A140451 A054147 * A043012 A122120 A080952

Adjacent sequences: A233579 A233580 A233581 * A233583 A233584 A233585

Keyword

nonn,cofr

Author

Stanislav Sykora, Jan 02 2014

Status

approved