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URL: https://oeis.org/A040003

⇱ A040003 - OEIS

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A040003
Continued fraction for sqrt(6).
6
2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4
OFFSET
0,1
COMMENTS
Decimal expansion of 37/165. - Elmo R. Oliveira, Oct 01 2025
REFERENCES
Harold Davenport, The Higher Arithmetic, Cambridge University Press, 8th ed., 2008, pp. 84-85, 97.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.4 Powers and Roots, p. 143.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.
FORMULA
a(n-1) = gcd(2^n, 3^n+1) (empirical). - Michel Marcus, Sep 03 2020
G.f.: 2*(1 + x + x^2)/(1 - x^2). - Stefano Spezia, Jul 26 2025
E.g.f.: 2*(2*cosh(x) + sinh(x) - 1). - Elmo R. Oliveira, Oct 01 2025
EXAMPLE
2.449489742783178098197284074... = 2 + 1/(2 + 1/(4 + 1/(2 + 1/(4 + ...)))). - Harry J. Smith, Jun 01 2009
MAPLE
Digits := 100: convert(evalf(sqrt(N)), confrac, 90, 'cvgts'):
MATHEMATICA
ContinuedFraction[Sqrt[6], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
PROG
(PARI) { allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(6)); for (n=0, 20000, write("b040003.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009
CROSSREFS
Equals twice A040001.
Essentially the same as A010694.
Cf. A010464 (decimal expansion), A041006/A041007 (convergents), A248236 (Egyptian fraction).
Sequence in context: A054763 A100374 A045841 * A106469 A082508 A327730
KEYWORD
nonn,cofr,easy
STATUS
approved