A122461 Repetitions of even numbers four times.

0, 0, 0, 0, 2, 2, 2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 8, 8, 8, 8, 10, 10, 10, 10, 12, 12, 12, 12, 14, 14, 14, 14, 16, 16, 16, 16, 18, 18, 18, 18, 20, 20, 20, 20, 22, 22, 22, 22, 24, 24, 24, 24, 26, 26, 26, 26, 28, 28, 28, 28, 30, 30, 30, 30, 32, 32, 32, 32, 34, 34, 34, 34, 36, 36, 36, 36

Offset: 0

Author

Paolo P. Lava and Giorgio Balzarotti, Oct 20 2006

Keywords

• nonn
• easy

Comments

Number of roots of P(x) = 1 + x + x^2 + … + x^n in the right half-plane. - Michel Lagneau, Oct 30 2012

Links

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

Crossrefs

Cf. A002265, A092533, A180969.

Programs

Maple

with(numtheory): for n from 1 to 70 do:it:=0:y:=[fsolve(sum('x^i ', 'i'=0..n-1), x, complex)] : for m from 1 to nops(y) do : if Re(y[m]) > 0 then it:=it+1:else fi:od: printf(`%d, `,it):od: # Michel Lagneau, Oct 31 2012
A122461:=n->2*floor(n/4); seq(A122461(n), n=0..100); # Wesley Ivan Hurt, Dec 06 2013

Mathematica

Table[2 Floor[n/4], {n, 0, 100}] (* Wesley Ivan Hurt, Dec 06 2013 *)
Table[PadRight[{},4,2n],{n,0,20}]//Flatten (* or *) LinearRecurrence[ {1,0,0,1,-1},{0,0,0,0,2},80] (* Harvey P. Dale, Mar 15 2020 *)

Python

def A122461(n): return n>>1&-2 # Chai Wah Wu, Jan 30 2023

Formula

a(n) = (Sum_{k=0..n} (k+1)*cos((n-k)*Pi/2))+1/4*(2*cos(n*Pi/2)+1+(-1)^n)-2. - Paolo P. Lava, May 15 2007

a(n) = 2*A002265(n) = 2*A180969(2,n). [Adriano Caroli, Nov 25 2010, corrected by R. J. Mathar, Nov 26 2010]

G.f.: 2*x^4/(1-x-x^4+x^5). [Bruno Berselli, Oct 31 2012]

a(n) = (-3+(-1)^n+2*i^((n-1)*n)+2*n)/4, where i=sqrt(-1). [Bruno Berselli, Oct 31 2012]

a(n) = 2 * floor(n/4). - Wesley Ivan Hurt, Dec 06 2013

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