A007877 Period 4 zigzag sequence: repeat [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, 1, 2, 1, 0

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

Table of n, a(n) for n=0..84.

P. Liu, Efficient recognition of integer sequences, Master's Essay, University of Waterloo, Dec. 1994. (Annotated scanned copy)

Index entries for sequences related to Chebyshev polynomials.

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

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) - (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

a(n) = a(n-4) for n >= 4. - Wesley Ivan Hurt, Sep 07 2022

a(n) = n - 2*floor(n/4) - 2*floor((n+1)/4). - Ridouane Oudra, Jan 22 2024

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 *)
PadRight[{}, 100, {0, 1, 2, 1}] (* Harvey P. Dale, Oct 24 2023 *)

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
(Python)
def A007877(n): return (0, 1, 2, 1)[n&3] # Chai Wah Wu, Jan 26 2023

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