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A000035
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Period 2: repeat [0, 1]; a(n) = n mod 2; parity of n.
(Formerly M0001)
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681
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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;
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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
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EXAMPLE
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G.f. = x + x^3 + x^5 + x^7 + x^9 + x^11 + x^13 + x^15 + ... - _Michael Somos_, Feb 20 2024
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MAPLE
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[ seq(i mod 2, i=0..100) ];
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MATHEMATICA
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PadLeft[{}, 110, {0, 1}] (* _Harvey P. Dale_, Sep 25 2011 *)
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PROG
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(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
makelist(A000035(n), n, 0, 30); /* _Martin Ettl_, Nov 12 2012 */
(Scheme) (define (A000035 n) (mod n 2)) ;; For R6RS. Use modulo in older Schemes like MIT/GNU Scheme. - _Antti Karttunen_, Mar 21 2017
(Python)
def A000035(n): return n & 1 # _Chai Wah Wu_, May 25 2022
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CROSSREFS
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Cf. A053644 for most significant bit.
This is Guy Steele's sequence GS(1, 2) (see A135416).
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KEYWORD
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cons,core,easy,nonn,nice,mult
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AUTHOR
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_N. J. A. Sloane_
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STATUS
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approved
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