A001204 Continued fraction for e^2. (Formerly M4322 N1811)

7, 2, 1, 1, 3, 18, 5, 1, 1, 6, 30, 8, 1, 1, 9, 42, 11, 1, 1, 12, 54, 14, 1, 1, 15, 66, 17, 1, 1, 18, 78, 20, 1, 1, 21, 90, 23, 1, 1, 24, 102, 26, 1, 1, 27, 114, 29, 1, 1, 30, 126, 32, 1, 1, 33, 138, 35, 1, 1, 36, 150, 38, 1, 1, 39, 162, 41, 1, 1, 42, 174, 44, 1, 1, 45, 186, 47, 1, 1

Offset

0,1

Comments

Note that e^2 = 7 + 2/(5 + 1/(7 + 1/(9 + 1/(11 + ...)))) (follows from the fact that A004273 is the continued fraction expansion of tanh(1) = (e^2 - 1)/(e^2 + 1)). - Peter Bala, Jan 15 2022

References

O. Perron, Die Lehre von den Kettenbrüchen, 2nd ed., Teubner, Leipzig, 1929, p. 138.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Harry J. Smith, Table of n, a(n) for n = 0..20000

K. Matthews, Finding the continued fraction of e^(l/m)

G. Xiao, Contfrac

Index entries for continued fractions for constants

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 2, 0, 0, 0, 0, -1).

Formula

G.f.: (x^10 - x^8 - x^7 + x^6 + 4x^5 + 3x^4 + x^3 + x^2 + 2x + 7)/(x^5 - 1)^2. - Ralf Stephan, Mar 23 2003

For n > 0, a(5n) = 12n + 6, a(5n+1) = 3n + 2, a(5n+2) = a(5n+3) = 1 and a(5n+4) = 3n + 3. - Dean Hickerson, Mar 25 2003

Example

7.389056098930650227230427460... = 7 + 1/(2 + 1/(1 + 1/(1 + 1/(3 + ...)))).

Mathematica

ContinuedFraction[ E^2, 100]
LinearRecurrence[{0, 0, 0, 0, 2, 0, 0, 0, 0, -1}, {7, 2, 1, 1, 3, 18, 5, 1, 1, 6, 30}, 80] (* Harvey P. Dale, Dec 30 2023 *)

Prog

(PARI) contfrac(exp(2))
(PARI) allocatemem(932245000); default(realprecision, 95000); x=contfrac(exp(2)); for (n=1, 20001, write("b001204.txt", n-1, " ", x[n])); \\ Harry J. Smith, Apr 30 2009

Crossrefs

Cf. A003417, A004273, A005131, A058282, A072334.

Sequence in context: A010505 A363360 A020844 * A177969 A021585 A103713

Adjacent sequences: A001201 A001202 A001203 * A001205 A001206 A001207

Keyword

nonn,easy,cofr,nice

Author

N. J. A. Sloane

Extensions

More terms from Robert G. Wilson v, Dec 07 2000

Status

approved