|
| |
|
|
A000035
|
|
Period 2: (0, 1) repeated; a(n) = n mod 2.
(Formerly M0001)
|
|
148
| |
|
|
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; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,1
|
|
|
COMMENTS
| Least significant bit of n, lsb(n).
Also decimal expansion of 1/99.
a(n) = ABS(A134451(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 27 2007
Characteristic function of odd numbers: a(A005408(n))=1, a(A005843(n))=0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 29 2008]
A102370(n) modulo 2 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 04 2009]
Base b expansion of 1/(b^2-1) for any b>=2 is 0.0101... (A005563 has b^2-1). [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 27 2009]
Let A be the Hessenberg n by 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). [From Milan R. Janjic (agnus(AT)blic.net), Jan 24 2010]
Contribution from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 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
|
|
|
REFERENCES
| Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. K. Whitford, Binet's Formula Generalized, Fib. Quart., 15 (1977), pp. 21, 24, 29.
|
|
|
LINKS
| David Wasserman, Table of n, a(n) for n = 0..1000
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
Index entries for "core" sequences
Index to sequences with linear recurrences with constant coefficients, signature (0,1).
Index entries for characteristic functions
|
|
|
FORMULA
| a(n) = (1 - (-1)^n)/2. a(n) = n mod 2.
Multiplicative with a(p^e) = p%2. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
G.f.: x/(1-x^2). E.g.f.: sinh(x). a(n)=n mod 2. a(n)=1/2 - (-1)^n/2. - Paul Barry (pbarry(AT)wit.ie), 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 (reinhard.zumkeller(AT)gmail.com), Apr 19 2005
a(n)= [sin(n*Pi/2)]^2 = [cos(n*Pi/2 +/- Pi/2)]^2 with n>=0. - Paolo P. Lava (paoloplava(AT)gmail.com), Sep 20 2006
Dirichlet g.f. (1-1/2^s)*zeta(s). - R. J. Mathar, Mar 04 2011
|
|
|
MAPLE
| A000035 := n->n mod 2;
[ seq(i mod 2, i=0..100) ];
|
|
|
MATHEMATICA
| Nest[Flatten[ # /. {0 -> {0, 1}, 1 -> {0, 1}}] &, {0}, 7] (from 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 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2009]
PadLeft[{}, 110, {0, 1}] (* From Harvey P. Dale, Sep 25 2011 *)
|
|
|
PROG
| (PARI) a(n)=n%2
See link in A140080 for Fortran program.
(Haskell)
a000035 n = a000035_list !! n
a000035_list = cycle [0, 1] -- Reinhard Zumkeller, Jan 06 2012
|
|
|
CROSSREFS
| Ones complement of A059841. Cf. A053644 for most significant bit.
This is Guy Steele's sequence GS(1, 2) (see A135416).
Sequence in context: A173923 A125122 * A188510 A131734 A134452 A073445
Adjacent sequences: A000032 A000033 A000034 * A000036 A000037 A000038
|
|
|
KEYWORD
| core,easy,nonn,nice,mult
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
