A233584 Coefficients of the generalized continued fraction expansion sqrt(e) = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

1, 1, 1, 1, 5, 9, 17, 109, 260, 2909, 3072, 3310, 3678, 6715, 35175, 37269, 439792, 1400459, 1472451, 4643918, 5683171, 44850176, 62252861, 145631385, 154435765, 371056666, 1685980637, 11196453405, 14795372939

Offset: 1

Author

Stanislav Sykora, Jan 06 2014

Keywords

• nonn

Comments

For more details on Blazys' expansions, see A233582.

Compared with simple continued fraction expansion for sqrt(e), this sequence starts soon growing very rapidly.

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. A019774 (sqrt(e)), A058281 (simple continued fraction).

Cf. Blazys' expansions: A233582 (Pi), A233583, A233585, A233586, A233587 and Blazys' continued fractions: A233588, A233589, A233590, 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[Sqrt@E, 35] (* Robert G. Wilson v, May 22 2014 *)

PARI

bx(x, nmax)={local(c, v, k); \\ Blazys expansion function
v = vector(nmax); c = x; for(k=1, nmax, v[k] = floor(c); c = v[k]/(c-v[k]); ); return (v); }
bx(exp(1/2), 100) \\ Execution; use high real precision

Formula

sqrt(e) = 1+1/(1+1/(1+1/(1+1/(5+5/(9+9/(17+17/(109+...))))))).