A059841 Period 2: Repeat [1,0]. a(n) = 1 - (n mod 2); Characteristic function of even numbers.

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

Offset: 0

Author

Alford Arnold, Feb 25 2001

Keywords

• easy
• nonn
• changed

Comments

When viewed as a triangular array, the row sum values are 0 1 1 1 2 3 3 3 4 5 5 5 6 ... (A004525).

This is the r=0 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found.

Successive binomial transforms of this sequence: A011782, A007051, A007582, A081186, A081187, A081188, A081189, A081190, A060531, A081192.

Characteristic function of even numbers: a(A005843(n))=1, a(A005408(n))=0. - Reinhard Zumkeller, Sep 29 2008

This sequence is the Euler transformation of A185012. - Jason Kimberley, Oct 14 2011

a(n) is the parity of n+1. - Omar E. Pol, Jan 17 2012

Read as partial sequences, we get to A000975. - Jon Perry, Nov 11 2014

Elementary Cellular Automata rule 77 produces this sequence. See Wolfram, Weisstein and Index links below. - Robert Price, Jan 30 2016

Column k = 1 of A051159. - John Keith, Jun 28 2021

When read as a constant: decimal expansion of 10/99, binary expansion of 2/3. - Jason Bard, Aug 25 2025

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1, 0;
  1, 0, 1, 0;
  1, 0, 1, 0, 1;
  0, 1, 0, 1, 0, 1;
  0, 1, 0, 1, 0, 1, 0;
  1, 0, 1, 0, 1, 0, 1, 0;
  1, 0, 1, 0, 1, 0, 1, 0, 1;
  0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
  0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0;
  ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
• Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
• Atsuto Seko, Atsushi Togo, and Isao Tanaka, Group-theoretical high-order rotational invariants for structural representations: Application to linearized machine learning interatomic potential, arXiv:1901.02118 [physics.comp-ph], 2019.
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• S. Wolfram, A New Kind of Science
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata
• Index entries for linear recurrences with constant coefficients, signature (0,1).
• Index entries for characteristic functions

Crossrefs

One's complement of A000035 (essentially the same, but shifted once).

Cf. A033999 (first differences), A008619 (partial sums), A004525, A011782 (binomial transf.), A000975.

Characteristic function of multiples of g: A000007 (g=0), A000012 (g=1), this sequence (g=2), A079978 (g=3), A121262 (g=4), A079998 (g=5), A079979 (g=6), A082784 (g=7).

Programs

Haskell

a059841 n = (1 -) . (`mod` 2)
a059841_list = cycle [1,0]
-- Reinhard Zumkeller, May 05 2012, Dec 30 2011

Magma

[0^(n mod 2): n in  [0..100]]; // Vincenzo Librandi, Nov 09 2014

Maple

seq(1-modp(n,2), n=0..150); # Muniru A Asiru, Apr 05 2018

Mathematica

CoefficientList[Series[1/(1 - x^2), {x, 0, 104}], x] (* or *)
Array[1/2 + (-1)^#/2 &, 105, 0] (* Michael De Vlieger, Feb 19 2019 *)
Table[QBinomial[n, 1, -1], {n, 1, 74}] (* John Keith, Jun 28 2021 *)
PadRight[{},120,{1,0}] (* Harvey P. Dale, Mar 06 2023 *)

PARI

a(n)=(n+1)%2; \\ or 1-n%2 as in NAME.

PARI

A059841(n)=!bittest(n,0) \\ M. F. Hasler, Jan 13 2012

Python

def A059841(n): return 1 - (n & 1) # Chai Wah Wu, May 25 2022

Formula

a(n) = 1 - A000035(n). - M. F. Hasler, Jan 13 2012

From Paul Barry, Mar 11 2003: (Start)

G.f.: 1/(1-x^2).

E.g.f.: cosh(x).

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

a(n) = 1/2 + (-1)^n/2. (End)

Additive with a(p^e) = 1 if p = 2, 0 otherwise.

a(n) = Sum_{k=0..n} (-1)^k*A038137(n, k). - Philippe Deléham, Nov 30 2006

a(n) = Sum_{k=1..n} (-1)^(n-k) for n > 0. - William A. Tedeschi, Aug 05 2011

E.g.f.: cosh(x) = 1 + x^2/(Q(0) - x^2); Q(k) = 8k + 2 + x^2/(1 + (2k + 1)*(2k + 2)/Q(k + 1)); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2011

E.g.f.: cosh(x) = 1/2*Q(0); Q(k) = 1 + 1/(1 - x^2/(x^2 + (2k + 1)*(2k + 2)/Q(k + 1))); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2011

E.g.f.: cosh(x) = E(0)/(1-x) where E(k) = 1 - x/(1 - x/(x - (2*k+1)*(2*k+2)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Apr 05 2013

For the general case: the characteristic function of numbers that are not multiples of m is a(n) = floor((n-1)/m) - floor(n/m) + 1, m,n > 0. - Boris Putievskiy, May 08 2013

a(n) = A000035(n+1) = A008619(n) - A110654(n). - Wesley Ivan Hurt, Jul 20 2013

Extensions

Better definition from M. F. Hasler, Jan 13 2012

Reinhard Zumkeller's Sep 29 2008 description added as a secondary name by Antti Karttunen, May 03 2022