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 A010121 Continued fraction for sqrt(7). 14

%I

%S 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,

%T 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,

%U 4,1,1,1,4,1,1,1,4,1,1,1,4

%N Continued fraction for sqrt(7).

%C 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).

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

%C 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

%H Harry J. Smith, <a href="/A010121/b010121.txt">Table of n, a(n) for n = 0..20000</a>

%H C. Elsner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Elsner/elsner9.html">Series of Error Terms for Rational Approximations of Irrational Numbers </a>, J. Int. Seq. 14 (2011) # 11.1.4, example 5.

%H <a href="/index/Con#confC">Index entries for continued fractions for constants</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 1).

%F a(n) = (1/24)*(-11*(n mod 4) + 7*((n+1) mod 4) + 7*((n+2) mod 4) + 25*((n+3) mod 4)) - 2*(binomial(2*n,n) mod 2), with n >= 0. - _Paolo P. Lava_, Jun 11 2009

%F From _R. J. Mathar_, Jun 17 2009: (Start)

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

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

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

%t ContinuedFraction[Sqrt[7],300] (* _Vladimir Joseph Stephan Orlovsky_, Mar 04 2011 *)

%t 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 *)

%t PadRight[{2},120,{4,1,1,1}] (* _Harvey P. Dale_, Nov 30 2019 *)

%o (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

%Y Cf. A010465 (decimal expansion).

%K nonn,cofr,mult

%O 0,1

%A _N. J. A. Sloane_

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Last modified April 5 06:13 EDT 2020. Contains 333238 sequences. (Running on oeis4.)