A001204 Continued fraction for e^2.
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
Examples
7.389056098930650227230427460... = 7 + 1/(2 + 1/(1 + 1/(1 + 1/(3 + ...)))).
References
- Oskar 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
- Keith Matthews, Finding the continued fraction of e^(l/m).
- Gang 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).
Programs
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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 *) -
PARI
contfrac(exp(2))
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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
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
Sum_{n>=5} (-1)^(n+1)/a(n) = (8*sqrt(3)-3)*Pi/72 - 2*log(2)/3. - Amiram Eldar, May 04 2025
Extensions
More terms from Robert G. Wilson v, Dec 07 2000
Comments