A012265 x - x^3/3! + x^5/5! - ... + (-1)^n*x^(2n+1)/(2n+1)! has 2a(n)+1 real roots.

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, 8, 9, 8, 9, 10, 9, 10, 11, 10, 11, 10, 11, 12, 11, 12, 11, 12, 13, 12, 13, 14, 13, 14, 13, 14, 15, 14, 15, 16

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

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

References

• James Propp, posting to math-fun mailing list May 30 1997.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A012264.

Programs

Mathematica

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 *)

Extensions

More terms from James Sellers