A290561 a(n) = n + cos(n*Pi/2).

1, 1, 1, 3, 5, 5, 5, 7, 9, 9, 9, 11, 13, 13, 13, 15, 17, 17, 17, 19, 21, 21, 21, 23, 25, 25, 25, 27, 29, 29, 29, 31, 33, 33, 33, 35, 37, 37, 37, 39, 41, 41, 41, 43, 45, 45, 45, 47, 49, 49, 49, 51, 53, 53, 53, 55, 57, 57, 57, 59, 61, 61, 61, 63, 65, 65, 65

Offset: 0

Author

Jean-François Alcover and Paul Curtz, Aug 06 2017

Keywords

• nonn
• easy

Comments

a(n) divides A289296(n).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Crossrefs

Cf. A004524, A085046, A289296, A290562.

Programs

Maple

A290561:=n->n+cos(n*Pi/2): seq(A290561(n), n=0..150); # Wesley Ivan Hurt, Aug 06 2017

Mathematica

a[n_] := n + Cos[n*Pi/2]; Table[a[n], {n, 0, 60}]

PARI

a(n) = n + round(cos(n*Pi/2)); \\ Michel Marcus, Aug 06 2017

PARI

Vec((x^3 + x^2 - x + 1)/((x - 1)^2*(x^2 + 1)) + O(x^100)) \\ Colin Barker, Aug 06 2017

Formula

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

a(n) = n if n == 3 (mod 4), and a(n) = a(n-4) + 4 otherwise, for n>2.

a(n) = a(n+20) - 20.

a(n) = 2*A004524(n) + 1.

a(n) + A290562(n) = 2*n.

a(n) * A290562(n) = n^2 - cos(n*Pi/2)^2 = A085046(n) for n>0.

A290562(n) = -a(-n).

From Colin Barker, Aug 06 2017: (Start)

a(n) = ((-i)^n + i^n)/2 + n where i=sqrt(-1).

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>3. (End)