A040001 1 followed by {1, 2} repeated.

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

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.

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)

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

From Jianing Song, Nov 01 2022: (Start)

For n > 0, a(n) is the minimal gap of distinct numbers coprime to n. Proof: denote the minimal gap by b(n). For odd n we have A058026(n) > 0, hence b(n) = 1. For even n, since 1 and -1 are both coprime to n we have b(n) <= 2, and that b(n) >= 2 is obvious.

The maximal gap is given by A048669. (End)

Links

Harry J. Smith, Table of n, a(n) for n = 0..20000

Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).

M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.

Ashok Kumar Gupta and Ashok Kumar Mittal, Bifurcating continued fractions, arXiv:math/0002227 [math.GM] (2000).

Michael 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 entries for 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

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 + ...))))
G.f. = 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 *)
PadRight[{1}, 120, {2, 1}] (* Harvey P. Dale, Nov 26 2015 *)

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])); } \\ Harry J. Smith, Jun 01 2009
(Haskell)
a040001 0 = 1; a040001 n = 2 - mod n 2
a040001_list = 1 : cycle [1, 2]  -- Reinhard Zumkeller, Apr 16 2015

Crossrefs

Cf. A000034, A002194, A133566, A083329 (binomial Transf).

Apart from a(0) the same as A134451.

Sequence in context: A168361 A107393 A000034 * A134451 A229217 A347526

Adjacent sequences: A039998 A039999 A040000 * A040002 A040003 A040004

Keyword

nonn,cofr,easy,mult,frac

Author

N. J. A. Sloane

Status

approved