OFFSET
0,5
COMMENTS
Conjecture: number of roots of P(x) = x^n - x^(n-1) - x^(n-2) - ... - x - 1 in the right half-plane. - Michel Lagneau, Apr 09 2013
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
FORMULA
a(n) = (Sum_{k=0..n} (k+1)*cos((n-k)*Pi/2)) + (1/4)*(2*cos(n*Pi/2) + 1 + (-1)^n) - 1, with n >= 0.
a(n) = 1 + 2*floor(n/4) = 1 + 2*A002265(n). - R. J. Mathar, Jun 10 2007
G.f.: (1+x^4)/((-1+x)^2*(1+x)*(x^2+1)). - R. J. Mathar, Nov 18 2007
a(n) = -1 + Sum_{k=0..n} ((1/12)*(-5*(k mod 4) + ((k+1) mod 4) + ((k+2) mod 4) + 7*((k+3) mod 4))). - Paolo P. Lava, Aug 21 2009
a(n) = n - A083219(n). - Michel Lagneau, Apr 09 2013
a(n) = (2*n + 1 + 2*cos(n*Pi/2) + cos(n*Pi) + 2*sin(n*Pi/2))/4. - Wesley Ivan Hurt, Oct 02 2017
From Stefano Spezia, May 26 2021: (Start)
E.g.f.: (cos(x) + cosh(x) + sin(x) + x*(cosh(x) + sinh(x)))/2.
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 4. (End)
MATHEMATICA
Table[1 + 2 Floor[n/4], {n, 0, 100}] (* Bruno Berselli, Jul 26 2014 *)
CoefficientList[Series[(1 + x^4)/(-1 + x)^2/(1 + x)/(x^2 + 1), {x, 0, 100}], x] (* Vincenzo Librandi, Jul 26 2014 *)
PROG
(Magma) [1+2*Floor(n/4): n in [0..100]]; // Bruno Berselli, Jul 26 2014
(Magma) I:=[1, 1, 1, 1, 3, 3, 3, 3, 5]; [n le 9 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..100]]; // Vincenzo Librandi, Jul 25 2014
(Python)
def A129756(n): return (n>>1)|1 # Chai Wah Wu, Jan 31 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paolo P. Lava and Giorgio Balzarotti, May 15 2007
STATUS
approved