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

Author

Christopher Lam Cham Kee (Topher(AT)CyberDude.Com)

Keywords

• nonn
• easy
• mult

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 sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

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).

Programs

Magma

&cat [[0,1,2,1]^^25]; // Vincenzo Librandi, Dec 27 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 *)
PadRight[{},100,{0,1,2,1}] (* Harvey P. Dale, Oct 24 2023 *)

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

Python

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

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

E.g.f.: exp(x) - cos(x). - Stefano Spezia, Aug 04 2025

Extensions

Chebyshev comments from Wolfdieter Lang, Sep 10 2004