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A040001 1 followed by {1, 2} repeated. 36
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; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

For n > 0: denominators of row sums of the triangular enumeration of rational numbers A226314(n,k) / A054531(n,k), 1 <= k <= n; see A226555 for numerators. - Reinhard Zumkeller, Jun 10 2013

LINKS

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).

FORMULA

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

a(n) = sign(n) + ((n+1) mod 2) = 1 + sign(n) - (n mod 2). - Wesley Ivan Hurt, Dec 13 2013

EXAMPLE

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 + ...

MAPLE

Digits := 100: convert(evalf(sqrt(N)), confrac, 90, 'cvgts'):

MATHEMATICA

ContinuedFraction[Sqrt[3], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011*)

PROG

(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]

CROSSREFS

Cf. A133566, A002194.

Sequence in context: A228826 A168361 A000034 * A134451 A229217 A167965

Adjacent sequences:  A039998 A039999 A040000 * A040002 A040003 A040004

KEYWORD

nonn,cofr,easy,mult,frac

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified October 1 12:17 EDT 2014. Contains 247510 sequences.