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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)/....))). 12
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 (list; graph; refs; listen; history; text; internal format)
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
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
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) 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
KEYWORD
nonn,easy
AUTHOR
Stanislav Sykora, Jan 06 2014
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)