cp's OEIS Frontend

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

%I A000035 M0001 #255 Feb 16 2025 08:32:19
%S A000035 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,
%T A000035 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,
%U A000035 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
%N A000035 Period 2: repeat [0, 1]; a(n) = n mod 2; parity of n.
%C A000035 Least significant bit of n, lsb(n).
%C A000035 Also decimal expansion of 1/99.
%C A000035 Also the binary expansion of 1/3. - _Robert G. Wilson v_, Sep 01 2015
%C A000035 a(n) = A134451(n) mod 2. - _Reinhard Zumkeller_, Oct 27 2007 [Corrected by _Jianing Song_, Nov 22 2019]
%C A000035 Characteristic function of odd numbers: a(A005408(n)) = 1, a(A005843(n)) = 0. - _Reinhard Zumkeller_, Sep 29 2008
%C A000035 A102370(n) modulo 2. - _Philippe Deléham_, Apr 04 2009
%C A000035 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
%C A000035 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
%C A000035 From _R. J. Mathar_, Jul 15 2010: (Start)
%C A000035 The sequence is the principal Dirichlet character of the reduced residue system mod 2 or mod 4 or mod 8 or mod 16 ...
%C A000035 Associated Dirichlet L-functions are for example L(2,chi) = Sum_{n>=1} a(n)/n^2 == A111003,
%C A000035 or L(3,chi) = Sum_{n>=1} a(n)/n^3 = 1.05179979... = 7*A002117/8,
%C A000035 or L(4,chi) = Sum_{n>=1} a(n)/n^4 = 1.014678... = A092425/96. (End)
%C A000035 Also parity of the nonnegative integers A001477. - _Omar E. Pol_, Jan 17 2012
%C A000035 a(n) = (4/n), where (k/n) is the Kronecker symbol. See the Eric Weisstein link. - _Wolfdieter Lang_, May 28 2013
%C A000035 Also the inverse binomial transform of A131577. - _Paul Curtz_, Nov 16 2016 [an observation forwarded by _Jean-François Alcover_]
%C A000035 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
%C A000035 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
%D A000035 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000035 David Wasserman, <a href="/A000035/b000035.txt">Table of n, a(n) for n = 0..1000</a>
%H A000035 Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%H A000035 F. Javier de Vega, <a href="https://arxiv.org/abs/2003.13378">An extension of Furstenberg's theorem of the infinitude of primes</a>, arXiv:2003.13378 [math.NT], 2020.
%H A000035 Clark Kimberling, <a href="https://www.emis.de/journals/JIS/VOL22/Kimberling/kimb9.html">A Combinatorial Classification of Triangle Centers on the Line at Infinity</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
%H A000035 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A000035 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DirichletSeriesGeneratingFunction.html">Dirichlet Series Generating Function</a>
%H A000035 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KroneckerSymbol.html">Kronecker Symbol</a>
%H A000035 A. K. Whitford, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/15-1/whitford-a.pdf">Binet's Formula Generalized</a>, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29
%H A000035 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A000035 <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%H A000035 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%H A000035 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,1).
%F A000035 a(n) = (1 - (-1)^n)/2.
%F A000035 a(n) = n mod 2.
%F A000035 a(n) = 1 - a(n-1).
%F A000035 Multiplicative with a(p^e) = p mod 2. - _David W. Wilson_, Aug 01 2001
%F A000035 G.f.: x/(1-x^2). E.g.f.: sinh(x). - _Paul Barry_, Mar 11 2003
%F A000035 a(n) = (A000051(n) - A014551(n))/2. - Mario Catalani (mario.catalani(AT)unito.it), Aug 30 2003
%F A000035 a(n) = ceiling((-2)^(-n-1)). - _Reinhard Zumkeller_, Apr 19 2005
%F A000035 Dirichlet g.f.: (1-1/2^s)*zeta(s). - _R. J. Mathar_, Mar 04 2011
%F A000035 a(n) = ceiling(n/2) - floor(n/2). - _Arkadiusz Wesolowski_, Sep 16 2012
%F A000035 a(n) = ceiling( cos(Pi*(n-1))/2 ). - _Wesley Ivan Hurt_, Jun 16 2013
%F A000035 a(n) = floor((n-1)/2) - floor((n-2)/2). - _Mikael Aaltonen_, Feb 26 2015
%F A000035 Dirichlet g.f.: L(chi(2),s) with chi(2) the principal Dirichlet character modulo 2. - _Ralf Stephan_, Mar 27 2015
%F A000035 a(n) = 0^^n = 0^(0^(0...)) (n times), where we take 0^0 to be 1. - _Natan Arie Consigli_, May 02 2015
%F A000035 Euler transform and inverse Moebius transform of length 2 sequence [0, 1]. - _Michael Somos_, Feb 20 2024
%e A000035 G.f. = x + x^3 + x^5 + x^7 + x^9 + x^11 + x^13 + x^15 + ... - _Michael Somos_, Feb 20 2024
%p A000035 A000035 := n->n mod 2;
%p A000035 [ seq(i mod 2, i=0..100) ];
%t A000035 PadLeft[{},110,{0,1}] (* _Harvey P. Dale_, Sep 25 2011 *)
%o A000035 (PARI) a(n)=n%2;
%o A000035 (PARI) a(n)=direuler(p=1,100,if(p==2,1,1/(1-X)))[n] /* _Ralf Stephan_, Mar 27 2015 */
%o A000035 (Haskell)
%o A000035 a000035 n = n `mod` 2  -- _James Spahlinger_, Oct 08 2012
%o A000035 (Haskell)
%o A000035 a000035_list = cycle [0,1]  -- _Reinhard Zumkeller_, Jan 06 2012
%o A000035 (Maxima) A000035(n):=mod(n,2)$
%o A000035 makelist(A000035(n),n,0,30); /* _Martin Ettl_, Nov 12 2012 */
%o A000035 (Scheme) (define (A000035 n) (mod n 2)) ;; For R6RS. Use modulo in older Schemes like MIT/GNU Scheme. - _Antti Karttunen_, Mar 21 2017
%o A000035 (Python)
%o A000035 def A000035(n): return n & 1 # _Chai Wah Wu_, May 25 2022
%Y A000035 Ones complement of A059841.
%Y A000035 Cf. A053644 for most significant bit.
%Y A000035 This is Guy Steele's sequence GS(1, 2) (see A135416).
%Y A000035 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).
%Y A000035 Cf. A154955 (Mobius transform), A131577 (binomial transform).
%Y A000035 Cf. A111003 (Dgf at s=2), A233091 (Dgf at s=3), A300707 (Dgf at s=4).
%Y A000035 Parity of A005811.
%K A000035 cons,core,easy,nonn,nice,mult
%O A000035 0,1
%A A000035 _N. J. A. Sloane_