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

Links

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

C. Elsner, Series of Error Terms for Rational Approximations of Irrational Numbers , J. Int. Seq. 14 (2011) # 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

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. A010465 (decimal expansion).

Sequence in context: A352894 A122374 A261960 * A174726 A300239 A354364

Adjacent sequences: A010118 A010119 A010120 * A010122 A010123 A010124

Keyword

nonn,cofr,easy,mult

Author

N. J. A. Sloane

Status

approved