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Search: id:A040001
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| A040001 |
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1 followed by {1, 2} repeated. |
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+0
16
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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
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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.
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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
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
G. Xiao, Contfrac
Index entries for continued fractions for constants
Eric Weisstein's World of Mathematics, Theodorus's Constant
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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 (davidwwilson(AT)comcast.net), Aug 01, 2001.
G.f.: (1+x+x^2)/(1-x^2). E.g.f.: (3exp(x)-2exp(0)+exp(-x))/2. - Paul Barry (pbarry(AT)wit.ie), 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 (pbarry(AT)wit.ie), Jun 22 2007
Row sums of triangle A133566 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 16 2007
a(n)=3/2+(1/2)*(-1)^n-[C(2*n,n) mod 2], with n>=0 - Paolo P. Lava (ppl(AT)spl.at), 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 + ...)))) [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jun 01 2009]
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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PROGRAM
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(PARI) a(n)=2-(n==0)-(n%2)
(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 (hjsmithh(AT)sbcglobal.net), Jun 01 2009]
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CROSSREFS
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Cf. A133566.
Cf. A002194 Decimal expansion. [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jun 01 2009]
Sequence in context: A022927 A063435 A000034 this_sequence A134451 A160990 A160989
Adjacent sequences: A039998 A039999 A040000 this_sequence 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 (njas(AT)research.att.com).
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