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 A001204 Continued fraction for e^2. (Formerly M4322 N1811) 6
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified July 19 11:53 EDT 2024. Contains 374394 sequences. (Running on oeis4.)