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A059841 Period 2: Repeat (1,0). a(n) = 1 - (n mod 2). 144
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; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

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.

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

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) = (sin((n+1)*Pi/2))^2 = (cos(n*Pi/2))^2 with n >= 0. - Paolo P. Lava, Nov 17 2006

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

a(n) = A000034(n) - 1. - Jon Perry, Nov 14 2014

EXAMPLE

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

MATHEMATICA

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

PROG

(PARI) a(n)=(n+1)%2;

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

(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

CROSSREFS

One's complement of A000035. Cf. A004525, A011782, A033999.

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).

Cf. A000975, A008619 (partial sums).

Sequence in context: A015301 A016213 A015757 * A056594 A101455 A091337

Adjacent sequences:  A059838 A059839 A059840 * A059842 A059843 A059844

KEYWORD

easy,nonn

AUTHOR

Alford Arnold, Feb 25 2001

EXTENSIONS

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

STATUS

approved

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Last modified February 23 11:36 EST 2018. Contains 299579 sequences. (Running on oeis4.)