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A010121
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Continued fraction for sqrt(7).
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9
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| This is a basic member of a family of 4-periodic multiplicative sequences
with two pameters (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
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LINKS
| Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
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FORMULA
| a(n)=(1/24)*{-11*(n mod 4)+7*[(n+1) mod 4]+7*[(n+2) mod 4]+25*[(n+3) mod 4]}-2*[C(2*n,n) mod 2], with n>=0 [From Paolo P. Lava (paoloplava(AT)gmail.com), Jun 11 2009]
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. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 17 2009]
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
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EXAMPLE
| 2.645751311064590590501615753... A010465 = 2 + 1/(1 + 1/(1 + 1/(1 + 1/(4 + ...))))
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MATHEMATICA
| ContinuedFraction[Sqrt[7], 300] (*From Vladimir Joseph Stephan Orlovsky, Mar 04 2011*)
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PROG
| (PARI) { allocatemem(932245000); default(realprecision, 13000); x=contfrac(sqrt(7)); for (n=0, 20000, write("b010121.txt", n, " ", x[n+1])); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jun 01 2009]
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CROSSREFS
| Sequence in context: A067856 A160467 A122374 * A174726 A205552 A157114
Adjacent sequences: A010118 A010119 A010120 * A010122 A010123 A010124
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KEYWORD
| nonn,cofr,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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