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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

• 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).

Crossrefs

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

Programs

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 *)

PARI

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

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