0,1

Binary representation of the Kempner-Mahler number Sum_{k>=0} 1/2^(2^k) = A007404.

Also a(n) = A(n) mod 2 where A is any of A001700, A005573, A007854, A026641, A049027, A064063, A064088, A064090, A064092, A064325, A064327, A064329, A064331, A064613, A076026, A105523, A123273, A126694, A126930, A126931, A126982, A126983, A126987, A127016, A127053, A127358, A127360, A127361, A127363. - Philippe Deléham, May 26 2007

a(n) = (product of digits of n; n in binary notation) mod 2. This sequence is a transformation of the Thue-Morse sequence (A010060), since there exists a function f such that f(sum of digits of n) = (product of digits of n). - Ctibor O. Zizka, Feb 12 2008

a(n-1), n >= 1, the characteristic sequence for powers of 2, A000079, is the unique solution of the following formal product and formal power series identity: Product_{j>=1} (1 + a(j-1)*x^j) = 1 + Sum_{k>=1} x^k = 1/(1-x). The product is therefore Product_{l>=1} (1 + x^(2^l)). Proof. Compare coefficients of x^n and use the binary representation of n. Uniqueness follows from the recurrence relation given for the general case under A147542. - Wolfdieter Lang, Mar 05 2009

a(n) is also the number of orbits of length n for the map x -> 1-cx^2 on [-1,1] at the Feigenbaum critical value c=1.401155... . - Thomas Ward, Apr 08 2009

A054525 (Mobius transform) * A001511 = A036987 = A047999^(-1) * A001511 = the inverse of Sierpiński's gasket * the ruler sequence. - Gary W. Adamson, Oct 26 2009 [Of course this is only vaguely correct depending on how the fuzzy indexing in these formulas is made concrete. - R. J. Mathar, Jun 20 2014]

Characteristic function of A000225. - Reinhard Zumkeller, Mar 06 2012

Also parity of the Catalan numbers A000108. - Omar E. Pol, Jan 17 2012

For n >= 2, also the largest exponent k >= 0 such that n^k in binary notation does not contain both 0 and 1. Unlike for the decimal version of this sequence, A062518, where the terms are only conjectural, for this sequence the values of a(n) can be proved to be the characteristic function of A000225, as follows: n^k will contain both 0 and 1 unless n^k = 2^r-1 for some r. But this is a special case of Catalan's equation x^p = y^q-1, which was proved by Preda Mihăilescu to have no nontrivial solution except 2^3 = 3^2 - 1. - Christopher J. Smyth, Aug 22 2014

Image, under the coding a,b -> 1; c -> 0, of the fixed point, starting with a, of the morphism a -> ab, b -> cb, c -> cc. - Jeffrey Shallit, May 14 2016

Number of nonisomorphic Boolean algebras of order n+1. - Jianing Song, Jan 23 2020

Antti Karttunen, Table of n, a(n) for n = 0..65537

D. Bailey et al., On the binary expansions of algebraic numbers, Journal de Théorie des Nombres de Bordeaux 16 (2004), 487-518.

Paul Barry, Some observations on the Rueppel sequence and associated Hankel determinants, arXiv:2005.04066 [math.CO], 2020.

Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.

Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.

Mats Granvik, Mathematica program for computing Fredholm Rueppel sequences

D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions, in Sequences and their Applications, C. Ding, T. Helleseth, and H. Niederreiter, eds., Proceedings of SETA'98 (Singapore, 1998), 308-317, 1999.

Preda Mihăilescu, Primary Cyclotomic Units and a Proof of Catalan's Conjecture, J. Reine angew. Math. 572 (2004): 167-195. doi:10.1515/crll.2004.048. MR 2076124.

H. Niederreiter and M. Vielhaber, Tree Complexity and a Doubly Exponential Gap between Structured and Random Sequences, J. Complexity, 12 (1996), 187-198.

Apisit Pakapongpun and Thomas Ward, Functorial orbit counting, Journal of Integer Sequences, 12 (2009) Article 09.2.4. [From Thomas Ward, Apr 08 2009]

Eric Rowland and Reem Yassawi, Profinite automata, arXiv:1403.7659 [math.DS], 2014. See p. 8.

E. Sheppard, net.math post (1985)

Stephen Wolfram, [Page 1092] A New Kind of Science | Online.

Index entries for characteristic functions

1 followed by a string of 2^k - 1 0's. Also a(n)=1 iff n = 2^m - 1.

a(n) = a(floor(n/2)) * (n mod 2) for n>0 with a(0)=1. - Reinhard Zumkeller, Aug 02 2002 [Corrected by Mikhail Kurkov, Jul 16 2019]

Sum_{n>=0} 1/10^(2^n) = 0.110100010000000100000000000000010...

1 if n=0, floor(log_2(n+1)) - floor(log_2(n)) otherwise. G.f.: (1/x) * Sum_{k>=0} x^(2^k) = Sum_{k>=0} x^(2^k-1). - Ralf Stephan, Apr 28 2003

a(n) = 1 - A043545(n). - Michael Somos, Aug 25 2003

a(n) = -Sum_{d|n+1} mu(2*d). - Benoit Cloitre, Oct 24 2003

Dirichlet g.f. for right-shifted sequence: 2^(-s)/(1-2^(-s)).

a(n) = A000108(n) mod 2 = A001405(n) mod 2. - Paul Barry, Nov 22 2004

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*Sum_{j=0..k} binomial(k, 2^j-1). - Paul Barry, Jun 01 2006

A000523(n+1) = Sum_{k=1..n} a(k). - Mitch Harris, Jul 22 2011

a(n) = A209229(n+1). - Reinhard Zumkeller, Mar 07 2012

a(n) = Sum_{k=1..n} A191898(n,k)*cos(Pi*(n-1)*(k-1))/n; (conjecture). - Mats Granvik, Mar 04 2013

a(n) = A000035(A000108(n)). - Omar E. Pol, Aug 06 2013

a(n) = 1 iff n=2^k-1 for some k, 0 otherwise. - M. F. Hasler, Jun 20 2014

a(n) = ceiling(log_2(n+2)) - ceiling(log_2(n+1)). - Gionata Neri, Sep 06 2015

From John M. Campbell, Jul 21 2016: (Start)

a(n) = (A000168(n-1) mod 2).

a(n) = (A000531(n+1) mod 2).

a(n) = (A000699(n+1) mod 2).

a(n) = (A000891(n) mod 2).

a(n) = (A000913(n-1) mod 2), for n>1.

a(n) = (A000917(n-1) mod 2), for n>0.

a(n) = (A001142(n) mod 2).

a(n) = (A001246(n) mod 2).

a(n) = (A001246(n) mod 4).

a(n) = (A002057(n-2) mod 2), for n>1.

a(n) = (A002430(n+1) mod 2). (End)

a(n) = 2 - A043529(n). - Antti Karttunen, Nov 19 2017

a(n) = floor(1+log(n+1)/log(2)) - floor(log(2n+1)/log(2)). - Adriano Caroli, Sep 22 2019

This is also the decimal expansion of -Sum_{k>=1} mu(2*k)/(10^k - 1), where mu is the Möbius function (A008683). - Amiram Eldar, Jul 12 2020

G.f. = 1 + x + x^3 + x^7 + x^15 + x^31 + x^63 + x^127 + x^255 + x^511 + ...

a(7) = 1 since 7 = 2^3 - 1, while a(10) = 0 since 10 is not of the form 2^k - 1 for any integer k.

A036987:= n-> `if`(2^ilog2(n+1) = n+1, 1, 0):

seq (A036987(n), n=0..128);

RealDigits[ N[ Sum[1/10^(2^n), {n, 0, Infinity}], 110]][[1]]

(* Recurrence: *)

t[n_, 1] = 1; t[1, k_] = 1;

t[n_, k_] := t[n, k] =

  If[n < k, If[n > 1 && k > 1, -Sum[t[k - i, n], {i, 1, n - 1}], 0],

   If[n > 1 && k > 1, Sum[t[n - i, k], {i, 1, k - 1}], 0]];

Table[t[n, k], {k, n, n}, {n, 104}]

(* Mats Granvik, Jun 03 2011 *)

mb2d[n_]:=1 - Module[{n2 = IntegerDigits[n, 2]}, Max[n2] - Min[n2]]; Array[mb2d, 120, 0] (* Vincenzo Librandi, Jul 19 2019 *)

Table[PadRight[{1}, 2^k, 0], {k, 0, 7}]//Flatten (* Harvey P. Dale, Apr 23 2022 *)

(PARI) {a(n) =( n++) == 2^valuation(n, 2)}; /* Michael Somos, Aug 25 2003 */

(Haskell)

a036987 n = ibp (n+1) where

   ibp 1 = 1

   ibp n = if r > 0 then 0 else ibp n' where (n', r) = divMod n 2

a036987_list = 1 : f [0, 1] where f (x:y:xs) = y : f (x:xs ++ [x, x+y])

-- Same list generator function as for a091090_list, cf. A091090.

-- Reinhard Zumkeller, May 19 2015, Apr 13 2013, Mar 13 2013

(Python)

from sympy import catalan

def a(n): return catalan(n)%2 # Indranil Ghosh, May 25 2017

Cf. A007404, A043545, A062518, A078885, A078585, A078886, A078887, A078888, A078889, A078890.

The first row of A073346. Occurs for first time in A073202 as row 6 (and again as row 8).

Congruent to any of the sequences A000108, A007460, A007461, A007463, A007464, A061922, A068068 reduced modulo 2. Characteristic function of A000225.

If interpreted with offset=1 instead of 0 (i.e., a(1)=1, a(2)=1, a(3)=0, a(4)=1, ...) then this is the characteristic function of 2^n (A000079) and as such occurs as the first row of A073265. Also, in that case the INVERT transform will produce A023359.

This is Guy Steele's sequence GS(1, 3), also GS(3, 1) (see A135416).

Cf. A054525, A047999. - Gary W. Adamson, Oct 26 2009

Cf. A043529, A127802.

Sequence in context: A262683 A321083 A154269 * A354193 A354188 A342025

Adjacent sequences:  A036984 A036985 A036986 * A036988 A036989 A036990

nonn

N. J. A. Sloane

Edited by M. F. Hasler, Jun 20 2014

approved