Search: id:A040001
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%I A040001
%S A040001 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,
%T A040001 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,
%U A040001 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
%N A040001 1 followed by {1, 2} repeated.
%C A040001 Continued fraction for sqrt(3).
%C A040001 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
%C A040001 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.
%H A040001 Harry J. Smith, <a href="b040001.txt">Table of n, a(n) for n=0,...,20000</
               a>
%H A040001 M. Beattie, S. D\u{a}sc\u{a}lescu and S. Raianu, <a href="http://front.math.ucdavis.edu/
               math.QA/0204075">Lifting of Nichols Algebras of Type B_2</a>
%H A040001 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SquareRoot.html">Link to a section of The World of Mathematics.</
               a>
%H A040001 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">
               Contfrac</a>
%H A040001 <a href="Sindx_Con.html#confC">Index entries for continued fractions 
               for constants</a>
%H A040001 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TheodorussConstant.html">Theodorus's Constant</a>
%F A040001 Multiplicative with a(p^e) = 2 if p even; 1 if p odd. - David W. Wilson 
               (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A040001 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
%F A040001 a(n)=(3-2*0^n +(-1)^n)/2. a(-n)=a(n). a(2n+1)=1, a(2n)=2, n nonzero.
%F A040001 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
%F A040001 Row sums of triangle A133566 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Sep 16 2007
%F A040001 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
%F A040001 Euler transform of length 3 sequence [ 1, 1, -1]. - Michael Somos Aug 
               04 2009
%F A040001 Moebius transform is length 2 sequence [ 1, 1]. - Michael Somos Aug 04 
               2009
%e A040001 1.732050807568877293527446341... = 1 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + ...)))) 
               [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jun 01 2009]
%e A040001 1 + x + 2*x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + x^9 + ...
%p A040001 Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):
%o A040001 (PARI) a(n)=2-(n==0)-(n%2)
%o A040001 (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]
%Y A040001 Cf. A133566.
%Y A040001 Cf. A002194 Decimal expansion. [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               Jun 01 2009]
%Y A040001 Sequence in context: A022927 A063435 A000034 this_sequence A134451 A160990 
               A160989
%Y A040001 Adjacent sequences: A039998 A039999 A040000 this_sequence A040002 A040003 
               A040004
%K A040001 nonn,cofr,easy,mult
%O A040001 0,3
%A A040001 N. J. A. Sloane (njas(AT)research.att.com).

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