A010121 Continued fraction for sqrt(7).

2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4

Offset

0,1

Comments

This is a basic member of a family of 4-periodic multiplicative sequences with two parameters (c1,c2), defined for n >= 1 by a(n)=1 if n is odd, a(n)=c1 if n == 0 (mod 4) and a(n)=c2 if n == 2 (mod 4). Here, (c1,c2)=(4,1).

The Dirichlet generating function is (1+(c2-1)/2^s+(c1-c2)/4^s)*zeta(s).

Other members are A010123 with parameters (6,2), A010127 (8,3), A010130 (10,1), A010131 (10,2), A010132 (10,4), A010137 (12,5), A010146 (14,6), A089146 (4,8), A109008 (4,2), A112132 (7,3). If c1=c2, this reduces to the cases discussed in A040001. - R. J. Mathar, Feb 18 2011

Decimal expansion of 10556/49995. - Elmo R. Oliveira, Oct 04 2025

References

Harold Davenport, The Higher Arithmetic, Cambridge University Press, 8th ed., 2008, p. 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

Carsten Elsner, Series of Error Terms for Rational Approximations of Irrational Numbers, J. Int. Seq., Vol. 14 (2011), Article 11.1.4, example 5.

G. Xiao, Contfrac.

Index entries for continued fractions for constants.

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

Formula

From R. J. Mathar, Jun 17 2009: (Start)

G.f.: -(2*x^2+3*x+2)*(x^2-x+1)/((x-1)*(1+x)*(x^2+1)).

a(n) = a(n-4), n > 4. (End)

a(n) = (7 + 3*(-1)^n + 3*(-i)^n + 3*i^n)/4, n > 0, where i is the imaginary unit. - Bruno Berselli, Feb 18 2011

E.g.f.: (3*cos(x) + 5*cosh(x) + 2*sinh(x) - 4)/2. - Elmo R. Oliveira, Oct 04 2025

Example

2.645751311064590590501615753...  = A010465 = 2 + 1/(1 + 1/(1 + 1/(1 + 1/(4 + ...)))).

Mathematica

ContinuedFraction[Sqrt[7], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
CoefficientList[Series[(2 x^2 + 3 x + 2) (x^2 - x + 1) / ((1 - x) (1 + x) (x^2 + 1)), {x, 0, 100}], x] (* Vincenzo Librandi, Nov 26 2016 *)
PadRight[{2}, 120, {4, 1, 1, 1}] (* Harvey P. Dale, Nov 30 2019 *)

Prog

(PARI) { allocatemem(932245000); default(realprecision, 13000); x=contfrac(sqrt(7)); for (n=0, 20000, write("b010121.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

Crossrefs

Cf. A040001.

Cf. A010123, A010127, A010130, A010131, A010132, A010137, A010146, A089146, A109008, A112132.

Cf. A010465 (decimal expansion), A041008/A041009 (convergents), A248237 (Egyptian fraction).

Sequence in context: A352894 A122374 A261960 * A392200 A174726 A300239

Adjacent sequences: A010118 A010119 A010120 * A010122 A010123 A010124

Keyword

nonn,cofr,easy,mult

Author

N. J. A. Sloane

Status

approved