A011658 Period 5: repeat [0, 0, 0, 1, 1]; also expansion of 1/(x^4 + x^3 + x^2 + x + 1) (mod 2).

0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0

Offset

0,1

Comments

Sequences of period k composed of (k-p) zeros followed by p ones have a closed formula of floor((n mod k)/(k-p)) for p >= floor(k/2). - Gary Detlefs, May 18 2011

a(n+3) is the determinant of an n X n pentadiagonal symmetric Toeplitz matrix with a=b=c=1. - R. J. Mathar, Jan 31 2023

Links

Table of n, a(n) for n=0..80.

J. Borowska and L. Lacinska, Recurrence form of determinant of a heptadiagonal symmetric Toeplitz matrix, J. Appl. Math. Comp. Mech. 13 (2014) 19-16, remark 1 for determinant of Toeplitz matrices a=b=c=1.

R. Gold, Characteristic linear sequences and their coset functions, J. SIAM Applied. Math., 14 (1966), 980-985.

Index entries for characteristic functions

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

Formula

a(n) = floor((n mod 5)/3). - Gary Detlefs, May 18 2011

a(n+2) = A198517(n+4) - A198517(n+2) + A198517(n). - Bruno Berselli, Nov 02 2011

a(n+4) = abs(a(n) - a(n+1) + a(n+2) - a(n+3)). - Benjamin Knight, May 06 2018

a(n) = (2/5) * (1 + cos(4*(n-4)*Pi/5) + cos(2*(n-3)*Pi/5) + cos(4*(n-3)*Pi/5) + cos(2*(n+1)*Pi/5)). - Wesley Ivan Hurt, Sep 26 2018

G.f.: -x^3*(1+x) / ( (x-1)*(1+x+x^2+x^3+x^4) ). - R. J. Mathar, Aug 11 2021

a(n) = floor(2*n/5) - 2*floor(n/5). - Ridouane Oudra, Apr 01 2023

Maple

seq(floor(2*n/5) - 2*floor(n/5), n=0..100); # Ridouane Oudra, Apr 01 2023

Mathematica

PadRight[{}, 120, {0, 0, 0, 1, 1}] (* Harvey P. Dale, Dec 16 2015 *)

Prog

(PARI) a(n)=(n%5)\3 \\ Charles R Greathouse IV, Jan 16 2017

Crossrefs

Cf. A198517. Parity of A010891(n+2).

Sequence in context: A288710 A179827 A285133 * A135461 A327219 A274950

Adjacent sequences: A011655 A011656 A011657 * A011659 A011660 A011661

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved