A012265 - OEIS

%I #20 Dec 16 2021 22:43:18

%S 0,1,0,1,2,1,2,3,2,3,2,3,4,3,4,3,4,5,4,5,6,5,6,5,6,7,6,7,8,7,8,7,8,9,

%T 8,9,8,9,10,9,10,11,10,11,10,11,12,11,12,11,12,13,12,13,14,13,14,13,

%U 14,15,14,15,16

%N x - x^3/3! + x^5/5! - ... + (-1)^n*x^(2n+1)/(2n+1)! has 2a(n)+1 real roots.

%C Let phi be the golden mean. Let B be the generalized Beatty sequence B(n):= 2*floor(n*phi) - 3*n, n = 0,1,2,... Then a(n) = B(n+5) for n = 0,...,200, except for n =  84, 118, 152, 165, 173, 186. - _Michel Dekking_, Mar 30 2020

%D _James Propp_, posting to math-fun mailing list May 30 1997.

%H Vincenzo Librandi, <a href="/A012265/b012265.txt">Table of n, a(n) for n = 0..200</a>

%t f[n_] := Sum[(-1)^k*x^(2*k + 1)/(2*k + 1)!, {k, 0, n}]; a[n_] := (CountRoots[f[n], x] - 1)/2; Table[a[n], {n, 0, 62}] (* _Jean-François Alcover_, Apr 16 2013 *)

%Y Cf. A012264.

%K nonn,easy,nice

%O 0,5

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_