A260686 Period 6 zigzag sequence, repeat [0, 1, 2, 3, 2, 1].

0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1

Offset: 0

Author

Wesley Ivan Hurt, Nov 15 2015

Keywords

• nonn
• easy

Comments

Decimal expansion of 37/3003. - Elmo R. Oliveira, Mar 06 2024

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Crossrefs

Period k zigzag sequences: A000035 (k=2), A007877 (k=4), this sequence (k=6), A266313 (k=8), A271751 (k=10), A271832 (k=12), A279313 (k=14), A279319 (k=16), A158289 (k=18).

Cf. A011655, A070435, A109007, A130151.

Programs

Magma

[1+(1-(-1)^n)/2-(-1)^Floor((n+1)/3): n in [0..100]]; // Bruno Berselli, Nov 16 2015

Magma

&cat[[0,1,2,3,2,1]: n in [0..15]]; // Vincenzo Librandi, Nov 17 2015

Maple

A260686:=n->[0, 1, 2, 3, 2, 1][(n mod 6)+1]: seq(A260686(n), n=0..100);

Mathematica

CoefficientList[Series[(x + x^2 + x^3)/(1 - x + x^3 - x^4), {x, 0, 100}], x]
Table[1 + (1 - (-1)^n)/2 - (-1)^Floor[(n + 1)/3], {n, 0, 100}] (* Bruno Berselli, Nov 16 2015 *)
PadRight[{}, 120, {0, 1, 2, 3, 2, 1}] (* Vincenzo Librandi, Nov 17 2015 *)

PARI

concat(0, Vec((x+x^2+x^3)/(1-x+x^3-x^4) + O(x^100))) \\ Altug Alkan, Nov 15 2015

Formula

G.f.: x*(1 + x + x^2) / (1 - x + x^3 - x^4).

a(n) = a(n-1) - a(n-3) + a(n-4) for n > 3.

a(n) = Sum_{i=1..n} (-1)^floor((i-1)/3) for n > 0.

a(n+1) = a(n) + A130151(n).

a(2n) = 2*A011655(n), a(2n+1) = A109007(n+2).

a(n) = 1 + (1 - (-1)^n)/2 - (-1)^floor((n+1)/3). - Bruno Berselli, Nov 16 2015

a(n) = sin(n*Pi/6)^2*(11+4*cos(n*Pi/3)+2*cos(2*n*Pi/3))/3. - Wesley Ivan Hurt, Jun 17 2016

a(n) = a(n-6) for n >= 6. - Wesley Ivan Hurt, Sep 07 2022

a(n) = sqrt(n^2 mod 12) = sqrt(A070435(n)). - Nicolas Bělohoubek, May 24 2024