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 A007877 Period 4 zigzag sequence: repeat [0,1,2,1]. 23
 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Euler transform of finite sequence [2,-2,0,1]. - Michael Somos, Sep 17 2004 This is the r = 2 member in the r-family of sequences S_r(n) defined in A092184 where more information can be found. a(n+1) is the transform of sqrt(1+2x)/sqrt(1-2x) (A063886) under the Chebyshev transformation A(x) -> (1/(1 + x^2))A(x/(1 + x^2)). See also A084099. - Paul Barry, Oct 12 2004 Multiplicative with a(2) = 2, a(2^e) = 0 if e >= 2, a(p^e) = 1 otherwise. - David W. Wilson, Jun 12 2005 The e.g.f. of 1, 2, 1, 0, 1, 2, 1, 0, ... (shifted left, offset zero) is exp(x) + sin(x). Binomial transform is A000749(n+2). - Wesley Ivan Hurt, Dec 30 2015 Decimal expansion of 11/909. - David A. Corneth, Dec 12 2016 Ternary expansion of 1/5. - J. Conrad, Aug 14 2017 LINKS P. Liu, Efficient recognition of integer sequences, Master's Essay, University of Waterloo, Dec. 1994. (Annotated scanned copy) Index entries for linear recurrences with constant coefficients, signature (1,-1,1). FORMULA Multiplicative with a(p^e) = 2 if p = 2 and e = 0; 0 if p = 2 and e > 0; 1 if p > 2. - David W. Wilson, Aug 01 2001 a(n) = -Sum_{k=0..n} (-1)^C(k+2, 2) (Offset -1). - Paul Barry, Jul 07 2003 a(n) = 1 - cos(n*Pi/2); a(n) = a(n-1) - a(n-2) + a(n-3) for n>2. - Lee Reeves (leereeves(AT)fastmail.fm), May 10 2004 a(n) = -a(n-2) + 2, n >= 2, a(0) = 0, a(1) = 1. G.f.: x*(1+x)/((1-x)*(1+x^2)) = x*(1+x)/(1-x+x^2-x^3). a(n) = 1 - T(n, 0) = 1 - A056594(n) with Chebyshev's polynomials T(n, x) of the first kind. Note that T(n, 0) = S(n, 0). a(n) = b(n) + b(n-1), n >= 1, with b(n) := A021913(n+1) the partial sums of S(n,0) = U(n,0) = A056594(n) (Chebyshev's polynomials evaluated at x=0). a(n) = 1 + (1/2){(-1)^[(n-1)/2] - (-1)^[n/2]}. - Ralf Stephan, Jun 09 2005 a(n) = 1/12*{5*(n mod 4) + 5*[(n+1) mod 4] - [(n+2) mod 4] - [(n+3) mod 4]}. - Paolo P. Lava, Oct 20 2006 Non-reduced g.f.: x*(1+x)^2/(1-x^4). - Jaume Oliver Lafont, Mar 27 2009 a(n+1) = (S(n, sqrt(2))^2, n >= 0, with the Chebyshev S-polynomials A049310. See the W. Lang link under A181878. - Wolfdieter Lang, Dec 15 2010 Dirichlet g.f. (1 + 1/2^s - 2/4^s)*zeta(s). - R. J. Mathar, Feb 24 2011 a(n) = n mod 4 - 2*(((n-2 mod 4) mod 3) mod 2). - Paolo P. Lava, Mar 13 2011 a(n) = (n mod 4) - (n^3 mod 4) + (n^2 mod 4). - Gary Detlefs, Apr 17 2011] a(n) = (n mod 2) + 2*floor(((n+1) mod 4)/3). - Gary Detlefs, Jul 19 2011] a(n) = sqrt(n^2 mod 8). - Wesley Ivan Hurt, Jan 01 2014 a(n) = (n AND 4*k+2)-(n AND 4*k+1) + 2*floor(((n+2) mod 4)/3), for any k. - Gary Detlefs, Jun 08 2014 a(n) = Sum_{i=1..n} (-1)^floor((i-1)/2). - Wesley Ivan Hurt, Dec 26 2015 MAPLE A007877:=n->sqrt(n^2 mod 8); seq(A007877(n), n=0..100); # Wesley Ivan Hurt, Jan 01 2014 MATHEMATICA f[n_] := Mod[n, 4] - Mod[n^3, 4] + Mod[n^2, 4] (* Or *) f[n_] := Mod[n, 2] + 2 Floor[Mod[n + 1, 4]/3] (* Or *) f[n_] := Switch[Mod[n, 4], 0, 0, 1, 1, 2, 2, 3, 1]; Array[f, 105, 0] (* Robert G. Wilson v, Aug 08 2011 *) Table[Sqrt[Mod[n^2, 8]], {n, 0, 100}] (* Wesley Ivan Hurt, Jan 01 2014 *) LinearRecurrence[{1, -1, 1}, {0, 1, 2}, 80] (* Vincenzo Librandi, Dec 27 2015 *) PROG (PARI) a(n)=[0, 1, 2, 1][1+n%4] \\ Jaume Oliver Lafont, Mar 27 2009 (PARI) concat(0, Vec(x*(1+x)/(1-x+x^2-x^3) + O(x^100))) \\ Altug Alkan, Dec 29 2015 (MAGMA) &cat [[0, 1, 2, 1]^^25]; // Vincenzo Librandi, Dec 27 2015 CROSSREFS Cf. A000749, A021913, A049310, A056594, A063886, A084099, A092184, A181878. Period k zigzag sequences: A000035 (k=2), this sequence (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: A199339 A323202 A118825 * A098178 A118822 A230074 Adjacent sequences:  A007874 A007875 A007876 * A007878 A007879 A007880 KEYWORD nonn,easy,mult AUTHOR Christopher Lam Cham Kee (Topher(AT)CyberDude.Com) EXTENSIONS Chebyshev comments from Wolfdieter Lang, Sep 10 2004 STATUS approved

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Last modified October 15 15:14 EDT 2019. Contains 328030 sequences. (Running on oeis4.)