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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)/....))). 11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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

FORMULA

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

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 *)

PROG

(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

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.

Sequence in context: A146067 A262484 A228956 * A262315 A061502 A110349

Adjacent sequences:  A233581 A233582 A233583 * A233585 A233586 A233587

KEYWORD

nonn

AUTHOR

Stanislav Sykora, Jan 06 2014

STATUS

approved

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Last modified June 28 16:56 EDT 2017. Contains 288839 sequences.