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

Author

Stanislav Sykora, Jan 06 2014

Keywords

• nonn
• easy

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

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.

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

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

Formula

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