A000035 Period 2: repeat [0, 1]; a(n) = n mod 2; parity of n.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• cons
• core
• easy
• nonn
• nice
• mult

Comments

Least significant bit of n, lsb(n).

Also decimal expansion of 1/99.

Also the binary expansion of 1/3. - Robert G. Wilson v, Sep 01 2015

a(n) = A134451(n) mod 2. - Reinhard Zumkeller, Oct 27 2007 [Corrected by Jianing Song, Nov 22 2019]

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

A102370(n) modulo 2. - Philippe Deléham, Apr 04 2009

Base b expansion of 1/(b^2-1) for any b >= 2 is 0.0101... (A005563 has b^2-1). - Rick L. Shepherd, Sep 27 2009

Let A be the Hessenberg n X n matrix defined by: A[1,j] = j mod 2, A[i,i] := 1, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = (-1)^n*charpoly(A,1). - Milan Janjic, Jan 24 2010

From R. J. Mathar, Jul 15 2010: (Start)

The sequence is the principal Dirichlet character of the reduced residue system mod 2 or mod 4 or mod 8 or mod 16 ...

Associated Dirichlet L-functions are for example L(2,chi) = Sum_{n>=1} a(n)/n^2 == A111003,

or L(3,chi) = Sum_{n>=1} a(n)/n^3 = 1.05179979... = 7*A002117/8,

or L(4,chi) = Sum_{n>=1} a(n)/n^4 = 1.014678... = A092425/96. (End)

Also parity of the nonnegative integers A001477. - Omar E. Pol, Jan 17 2012

a(n) = (4/n), where (k/n) is the Kronecker symbol. See the Eric Weisstein link. - Wolfdieter Lang, May 28 2013

Also the inverse binomial transform of A131577. - Paul Curtz, Nov 16 2016 [an observation forwarded by Jean-François Alcover]

The emanation sequence for the globe category. That is take the globe category, take the corresponding polynomial comonad, consider its carrier polynomial as a generating function, and take the corresponding sequence. - David Spivak, Sep 25 2020

For n > 0, a(n) is the alternating sum of the product of n increasing and n decreasing odd factors. For example, a(4) = 1*7 - 3*5 + 5*3 - 7*1 and a(5) = 1*9 - 3*7 + 5*5 - 7*3 + 9*1. - Charlie Marion, Mar 24 2022

Examples

			G.f. = x + x^3 + x^5 + x^7 + x^9 + x^11 + x^13 + x^15 + ... - _Michael Somos_, Feb 20 2024
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• David Wasserman, 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.
• F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
• Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
• Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
• Eric Weisstein's World of Mathematics, Dirichlet Series Generating Function
• Eric Weisstein's World of Mathematics, Kronecker Symbol
• A. K. Whitford, Binet's Formula Generalized, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29
• Index entries for "core" sequences
• Index entries for sequences that are fixed points of mappings
• Index entries for characteristic functions
• Index entries for linear recurrences with constant coefficients, signature (0,1).

Crossrefs

Ones complement of A059841.

Cf. A053644 for most significant bit.

This is Guy Steele's sequence GS(1, 2) (see A135416).

Period k zigzag sequences: this sequence (k=2), A007877 (k=4), A260686 (k=6), A266313 (k=8), A271751 (k=10), A271832 (k=12), A279313 (k=14), A279319 (k=16), A158289 (k=18).

Cf. A154955 (Mobius transform), A131577 (binomial transform).

Cf. A111003 (Dgf at s=2), A233091 (Dgf at s=3), A300707 (Dgf at s=4).

Parity of A005811.

Programs

Haskell

a000035 n = n `mod` 2  -- James Spahlinger, Oct 08 2012

Haskell

a000035_list = cycle [0,1]  -- Reinhard Zumkeller, Jan 06 2012

Maple

A000035 := n->n mod 2;
[ seq(i mod 2, i=0..100) ];

Mathematica

PadLeft[{},110,{0,1}] (* Harvey P. Dale, Sep 25 2011 *)

Maxima

A000035(n):=mod(n,2)$
makelist(A000035(n),n,0,30); /* Martin Ettl, Nov 12 2012 */

PARI

a(n)=n%2;

PARI

a(n)=direuler(p=1,100,if(p==2,1,1/(1-X)))[n] /* Ralf Stephan, Mar 27 2015 */

Python

def A000035(n): return n & 1 # Chai Wah Wu, May 25 2022

Scheme

(define (A000035 n) (mod n 2)) ;; For R6RS. Use modulo in older Schemes like MIT/GNU Scheme. - Antti Karttunen, Mar 21 2017

Formula

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

a(n) = n mod 2.

a(n) = 1 - a(n-1).

Multiplicative with a(p^e) = p mod 2. - David W. Wilson, Aug 01 2001

G.f.: x/(1-x^2). E.g.f.: sinh(x). - Paul Barry, Mar 11 2003

a(n) = (A000051(n) - A014551(n))/2. - Mario Catalani (mario.catalani(AT)unito.it), Aug 30 2003

a(n) = ceiling((-2)^(-n-1)). - Reinhard Zumkeller, Apr 19 2005

Dirichlet g.f.: (1-1/2^s)*zeta(s). - R. J. Mathar, Mar 04 2011

a(n) = ceiling(n/2) - floor(n/2). - Arkadiusz Wesolowski, Sep 16 2012

a(n) = ceiling( cos(Pi*(n-1))/2 ). - Wesley Ivan Hurt, Jun 16 2013

a(n) = floor((n-1)/2) - floor((n-2)/2). - Mikael Aaltonen, Feb 26 2015

Dirichlet g.f.: L(chi(2),s) with chi(2) the principal Dirichlet character modulo 2. - Ralf Stephan, Mar 27 2015

a(n) = 0^^n = 0^(0^(0...)) (n times), where we take 0^0 to be 1. - Natan Arie Consigli, May 02 2015

Euler transform and inverse Moebius transform of length 2 sequence [0, 1]. - Michael Somos, Feb 20 2024