A083219 a(n) = n - 2*floor(n/4).

0, 1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7, 8, 9, 8, 9, 10, 11, 10, 11, 12, 13, 12, 13, 14, 15, 14, 15, 16, 17, 16, 17, 18, 19, 18, 19, 20, 21, 20, 21, 22, 23, 22, 23, 24, 25, 24, 25, 26, 27, 26, 27, 28, 29, 28, 29, 30, 31, 30, 31, 32, 33, 32, 33, 34, 35, 34, 35, 36, 37, 36, 37, 38

Offset: 0

Author

Reinhard Zumkeller, Apr 22 2003

Keywords

• nonn
• easy
• changed

Comments

Conjecture: number of roots of P(x) = x^n - x^(n-1) - x^(n-2) - ... - x - 1 in the left half-plane. - Michel Lagneau, Apr 09 2013

a(n) is n+2 with its second least significant bit removed (see A021913(n+2) for that bit). - Kevin Ryde, Dec 13 2019

Links

• Altug Alkan, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A083220, A129756, A162751 (second highest bit removed).

Cf. A021913, A162330, A285869.

Essentially the same as A018837.

Programs

Maple

a:= n-> n - 2*floor(n/4):
seq(a(n), n=0..74);  # Alois P. Heinz, Jan 24 2021

Mathematica

Array[# - 2 Floor[#/4] &, 75, 0] (* Michael De Vlieger, Oct 02 2017 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,2,3,2},80] (* Harvey P. Dale, Oct 01 2018 *)

PARI

a(n) = n - n\4*2 \\ Charles R Greathouse IV, Jun 11 2015

Formula

a(n) = A083220(n)/2.

a(n) = a(n-1) + n mod 2 + (n mod 4 - 1)*(1 - n mod 2), a(0) = 0.

G.f.: x*(1+x+x^2-x^3)/((1-x)^2*(1+x)*(1+x^2)). - R. J. Mathar, Aug 28 2008

a(n) = n - A129756(n). - Michel Lagneau, Apr 09 2013

Bisection: a(2*k) = 2*floor((n+2)/4), a(2*k+1) = a(2*k) + 1, k >= 0. - Wolfdieter Lang, May 08 2017

a(n) = (2*n + 3 - 2*cos(n*Pi/2) - cos(n*Pi) - 2*sin(n*Pi/2))/4. - Wesley Ivan Hurt, Oct 02 2017

a(n) = A162330(n+2) - 1 = A285869(n+3) - 1. - Kevin Ryde, Dec 13 2019

E.g.f.: ((1 + x)*cosh(x) - cos(x) + (2 + x)*sinh(x) - sin(x))/2. - Stefano Spezia, May 27 2021

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2) - 1. - Amiram Eldar, Aug 21 2023