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A040001
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1 followed by {1, 2} repeated.
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34
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1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2
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OFFSET
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0,3
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COMMENTS
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Continued fraction for sqrt(3).
Also coefficient of the highest power of q in the expansion of the polynomial nu(n) defined by: nu(0)=1, nu(1)=b and for n>=2, nu(n)=b*nu(n-1)+lambda*(n-1)_q*nu(n-2) with (b,lambda)=(1,1), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
nu(0)=1 nu(1)=1; nu(2)=2; nu(3)=3+q; nu(4)=5+3q+2q^2; nu(5)=8+7q+6q^2+4q^3+q^4; nu(6)=13+15q+16q^2+14q^3+11q^4+5q^5+2q^6.
Contribution from Jaroslav Krizek, May 28 2010: (Start)
a(n) = denominators of arithmetic means of the first n positive integers for n >= 1.
See A026741(n+1) or A145051(n) - denominators of arithmetic means of the first n positive integers. (End)
Contribution from R. J. Mathar, Feb 16 2011: (Start)
This is a prototype of multiplicative sequences defined by a(p^e)=1 for
odd primes p, and a(2^e)=c with some constant c, here c=2.
They have Dirichlet generating functions (1+(c-1)/2^s)*zeta(s).
Examples are A153284, A176040 (c=3), A040005 (c=4), A021070, A176260 (c=5),
A040011, A176355 (c=6), A176415 (c=7), A040019, A021059 (c=8), A040029 (c=10), A040041 (c=12). (End)
a(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = A000325(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012
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LINKS
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Harry J. Smith, Table of n, a(n) for n = 0..20000
M. Beattie, S. D\u{a}sc\u{a}lescu and S. Raianu, Lifting of Nichols Algebras of Type B_2
M. Somos, Rational Function Multiplicative Coefficients
Eric Weisstein's World of Mathematics, Square Root
Eric Weisstein's World of Mathematics, Theodorus's Constant
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index to sequences with linear recurrences with constant coefficients, signature (0,1).
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FORMULA
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Multiplicative with a(p^e) = 2 if p even; 1 if p odd. - David W. Wilson, Aug 01, 2001.
G.f.: (1 + x + x^2) / (1 - x^2). E.g.f.: (3*exp(x)-2*exp(0)+exp(-x))/2. - Paul Barry, Apr 27 2003
a(n)=(3-2*0^n +(-1)^n)/2. a(-n)=a(n). a(2n+1)=1, a(2n)=2, n nonzero.
a(n)=sum{k=0..n, F(n-k+1)*(-2+(1+(-1)^k)/2+C(2, k)+0^k)}; - Paul Barry, Jun 22 2007
Row sums of triangle A133566 - Gary W. Adamson, Sep 16 2007
a(n)=3/2+(1/2)*(-1)^n-[C(2*n,n) mod 2], with n>=0 - Paolo P. Lava, Nov 27 2007
Euler transform of length 3 sequence [ 1, 1, -1]. - Michael Somos, Aug 04 2009
Moebius transform is length 2 sequence [ 1, 1]. - Michael Somos, Aug 04 2009
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EXAMPLE
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1.732050807568877293527446341... = 1 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + ...))))
1 + x + 2*x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + x^9 + ...
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MAPLE
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Digits := 100: convert(evalf(sqrt(N)), confrac, 90, 'cvgts'):
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MATHEMATICA
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ContinuedFraction[Sqrt[3], 300] (*From Vladimir Joseph Stephan Orlovsky, Mar 04 2011*)
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PROG
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(PARI) {a(n) = 2 - (n==0) - (n%2)} /* Michael Somos, Jun 11 2003 */
(PARI) { allocatemem(932245000); default(realprecision, 12000); x=contfrac(sqrt(3)); for (n=0, 20000, write("b040001.txt", n, " ", x[n+1])); } [From Harry J. Smith, Jun 01 2009]
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CROSSREFS
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Cf. A133566, A002194.
Sequence in context: A167964 A168361 A000034 * A134451 A167965 A167966
Adjacent sequences: A039998 A039999 A040000 * A040002 A040003 A040004
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KEYWORD
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nonn,cofr,easy,mult
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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