A129756 Repetitions of odd numbers four times.

1, 1, 1, 1, 3, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 9, 9, 9, 9, 11, 11, 11, 11, 13, 13, 13, 13, 15, 15, 15, 15, 17, 17, 17, 17, 19, 19, 19, 19, 21, 21, 21, 21, 23, 23, 23, 23, 25, 25, 25, 25, 27, 27, 27, 27, 29, 29, 29, 29, 31, 31, 31, 31, 33, 33, 33, 33, 35, 35, 35, 35, 37, 37, 37, 37

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

Cf. A002265, A083219.

Sequence in context: A111756 A130821 A352226 * A339971 A156724 A196186

Adjacent sequences: A129753 A129754 A129755 * A129757 A129758 A129759

Keyword

nonn,easy

Author

Paolo P. Lava and Giorgio Balzarotti, May 15 2007

Status

approved