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A000035 Period 2: repeat [0, 1]; a(n) = n mod 2; parity of n.
(Formerly M0001)
359
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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) = ABS(A134451(n)). - Reinhard Zumkeller, Oct 27 2007

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].

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.

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).

FORMULA

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

a(n) = n mod 2.

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

Multiplicative with a(p^e) = p%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) = ceil((-2)^(-n-1)). - Reinhard Zumkeller, Apr 19 2005

a(n) = (sin(n*Pi/2))^2 = (cos(n*Pi/2 +/- Pi/2))^2 with n >= 0. - Paolo P. Lava, Sep 20 2006

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

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

a(n) = ceil( 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

MAPLE

A000035 := n->n mod 2;

[ seq(i mod 2, i=0..100) ];

MATHEMATICA

Nest[Flatten[ # /. {0 -> {0, 1}, 1 -> {0, 1}}] &, {0}, 7] (* Robert G. Wilson v, Mar 05 2005 *)

Nest[ Flatten[ # /. {0 -> {0, 1, 0}}] &, {0}, 5] (* Robert G. Wilson v, Sep 01 2005 *)

CellularAutomaton[50, {{0, 1}, 0}, 104, {All, {0}}] // Flatten (* Zerinvary Lajos, Jul 08 2009 *)

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

a[n_] := Floor[(n-1)/2] - Floor[(n-2)/2]; Array[a, 105, 0] (* Robert G. Wilson v, Feb 26 2015 *)

PROG

(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 */

(Haskell)

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

(Haskell)

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

(Maxima) A000035(n):=mod(n, 2)$

makelist(A000035(n), n, 0, 30); /* Martin Ettl, Nov 12 2012 */

(MAGMA) [0^(1 - n mod 2): n in  [0..100]]; // Vincenzo Librandi, Nov 09 2014

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

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).

Sequence in context: A173923 A125122 * A188510 A131734 A134452 A073445

Adjacent sequences:  A000032 A000033 A000034 * A000036 A000037 A000038

KEYWORD

cons,core,easy,nonn,nice,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified June 25 00:41 EDT 2017. Contains 288708 sequences.