A030300 Runs have lengths 2^n, n >= 0.

1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Jean-Paul Delahaye

Keywords

• nonn
• base
• easy

Comments

An example of a sequence with property that the fraction of 1's in the first n terms does not converge to a limit. - N. J. A. Sloane, Sep 24 2007

Image, under the coding sending a,d,e -> 1 and b,c -> 0, of the fixed point, starting with a, of the morphism a -> ab, b -> cd, c -> ee, d -> eb, e -> cc. - Jeffrey Shallit, May 14 2016

This sequence taken as digits of a base-b fraction is g(1/b) = Sum_{n>=1} a(n)/b^n = b/(b-1) * Sum_{k>=0} (-1)^k/b^(2^k) per the generating function below. With initial 0, it is binary expansion .01001111 = A275975. With initial 0 and digits 2*a(n), it is ternary expansion .02002222 = A160386. These and in general g(1/b) for any integer b>=2 are among forms which Kempner showed are transcendental. - Kevin Ryde, Sep 07 2019

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Aubrey J. Kempner, On Transcendental Numbers, Transactions of the American Mathematical Society, volume 17, number 4, October 1916, pages 476-482.
• Kevin Ryde, Plot of A079947(n)/n, illustrating proportion of 1s in the first n terms here does not converge (but oscillates with rises and falls by hyperbolas)
• Ralf Stephan, Some divide-and-conquer sequences ...
• Ralf Stephan, Table of generating functions
• Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
• Index entries for sequences related to binary expansion of n
• Index entries for characteristic functions

Crossrefs

Cf. A030301. Partial sums give A079947.

Cf. A065359, A083905.

Characteristic function of A053738.

Programs

Maple

f0 := n->[seq(0,i=1..2^n)]; f1 := n->[seq(1,i=1..2^n)]; s := []; for i from 0 to 4 do s := [op(s), op(f1(2*i)), op(f0(2*i+1))]; od: A030300 := s;

Mathematica

nMax = 6; Table[1 - Mod[n, 2], {n, 0, nMax}, {2^n}] // Flatten (* Jean-François Alcover, Oct 20 2016 *)

PARI

a(n) = if(n, !(logint(n,2)%2)); /* Kevin Ryde, Aug 02 2019 */

Python

def A030300(n): return n.bit_length()&1 # Chai Wah Wu, Jan 30 2023

Formula

a(n) = A065359(n) + A083905(n).

a(n) = (1/2)*(1+(-1)^floor(log_2(n))). - Benoit Cloitre, Feb 22 2003

G.f.: 1/(1-x) * Sum_{k>=0} (-1)^k*x^2^k. - Ralf Stephan, Jul 12 2003

a(n) = 1 - a(floor(n/2)). - Vladeta Jovovic, Aug 04 2003

a(n) = A115253(2n, n) mod 2. - Paul Barry, Jan 18 2006

a(n) = 1 - A030301(n). - Antti Karttunen, Oct 10 2017