A233583 Coefficients of the generalized continued fraction expansion e = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

2, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56

Offset

1,1

Comments

For more details on Blazys' expansion, see A233582.

This sequence matches that of natural numbers (A000027), offset by 1, with two different starting terms.

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.

Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.

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

e = 2+2/(2+2/(2+2/(3+3/(4+4/(5+...))))).

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[E, 80] (* Robert G. Wilson v, May 22 2014 *)

Prog

(PARI)
default(realprecision, 100);
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), 100) \\ Execution; use high real precision

Crossrefs

Cf. A000027 (natural numbers), A001113 (number e).

Cf. Blazys' expansions: A233582 (Pi), A233584, A233585, A233586, A233587 and Blazys' continued fractions: A233588, A233589, A233590, A233591.

Sequence in context: A365718 A163801 A323735 * A309689 A029049 A094983

Adjacent sequences: A233580 A233581 A233582 * A233584 A233585 A233586

Keyword

nonn,easy

Author

Stanislav Sykora, Jan 06 2014

Status

approved