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

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy
• mult

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

Examples

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

References

• 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
• 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).

Crossrefs

Cf. A010465 (decimal expansion).

Programs

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 *)

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

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