|
%I #54 Feb 01 2023 02:30:40
%S 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,
%T 15,15,15,17,17,17,17,19,19,19,19,21,21,21,21,23,23,23,23,25,25,25,25,
%U 27,27,27,27,29,29,29,29,31,31,31,31,33,33,33,33,35,35,35,35,37,37,37,37
%N Repetitions of odd numbers four times.
%C 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
%H Vincenzo Librandi, <a href="/A129756/b129756.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F 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.
%F a(n) = 1 + 2*floor(n/4) = 1 + 2*A002265(n). - _R. J. Mathar_, Jun 10 2007
%F G.f.: (1+x^4)/((-1+x)^2*(1+x)*(x^2+1)). - _R. J. Mathar_, Nov 18 2007
%F 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
%F a(n) = n - A083219(n). - _Michel Lagneau_, Apr 09 2013
%F 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
%F From _Stefano Spezia_, May 26 2021: (Start)
%F E.g.f.: (cos(x) + cosh(x) + sin(x) + x*(cosh(x) + sinh(x)))/2.
%F a(n) = a(n-1) + a(n-4) - a(n-5) for n > 4. (End)
%t Table[1 + 2 Floor[n/4], {n, 0, 100}] (* _Bruno Berselli_, Jul 26 2014 *)
%t CoefficientList[Series[(1 + x^4)/(-1 + x)^2/(1 + x)/(x^2 + 1), {x, 0, 100}], x] (* _Vincenzo Librandi_, Jul 26 2014 *)
%o (Magma) [1+2*Floor(n/4): n in [0..100]]; // _Bruno Berselli_, Jul 26 2014
%o (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
%o (Python)
%o def A129756(n): return (n>>1)|1 # _Chai Wah Wu_, Jan 31 2023
%Y Cf. A002265, A083219.
%K nonn,easy
%O 0,5
%A _Paolo P. Lava_ and _Giorgio Balzarotti_, May 15 2007
|
