A040002 - OEIS

A040002

Continued fraction for sqrt(5).

2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Decimal expansion of 11/45. - Natan Arie Consigli, Jan 19 2016

REFERENCES

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.

Harold Davenport, The Higher Arithmetic, Cambridge University Press, 8th ed., 2008, pp. 84-85, 97.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

LINKS

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

G. Xiao, Contfrac

Index entries for continued fractions for constants

Index of eventually constant sequences

Index of divisibility sequences

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

FORMULA

a(0) = 2, a(n) = 4 n>0. - Natan Arie Consigli, Jan 19 2016

From Elmo R. Oliveira, Feb 16 2024: (Start)

G.f.: 2*(1+x)/(1-x).

E.g.f.: 4*exp(x) - 2.

a(n) = 2*A040000(n). (End)

EXAMPLE

2.236067977499789696409173668... = 2 + 1/(4 + 1/(4 + 1/(4 + 1/(4 + ...)))). - Harry J. Smith, Jun 01 2009

MAPLE

Digits := 100: convert(evalf(sqrt(N)), confrac, 90, 'cvgts'):

MATHEMATICA

ContinuedFraction[Sqrt[5], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)

PadRight[{2}, 120, {4}] (* Harvey P. Dale, Jul 06 2019 *)

PROG

(PARI) { allocatemem(932245000); default(realprecision, 26000); x=contfrac(sqrt(5)); for (n=0, 20000, write("b040002.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

CROSSREFS

Cf. A002163 (decimal expansion), A001077/A001076 (convergents), A248235 (Egyptian fraction).

Cf. Continued fraction for sqrt(a^2+1) = (a, 2a, 2a, 2a....): A040000 (contfrac(sqrt(2)) = (1,2,2,...)), A040002, A040006, A040012, A040020, A040030, A040042, A040056, A040072, A040090, A040110 (contfrac(sqrt(122)) = (11,22,22,...)), A040132, A040156, A040182, A040210, A040240, A040272, A040306, A040342, A040380, A040420 (contfrac(sqrt(442)) = (21,42,42,...)), A040462, A040506, A040552, A040600, A040650, A040702, A040756, A040812, A040870, A040930 (contfrac(sqrt(962)) = (31,62,62,...)).

Essentially the same as A010709.

Sequence in context: A105192 A345438 A203638 * A151798 A334897 A035684

Adjacent sequences: A039999 A040000 A040001 * A040003 A040004 A040005

KEYWORD

nonn,cofr,easy,cons

AUTHOR

N. J. A. Sloane

STATUS

approved