This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A012265 #21 Jul 02 2025 16:01:55
%S A012265 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 A012265 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 A012265 14,15,14,15,16
%N A012265 x - x^3/3! + x^5/5! - ... + (-1)^n*x^(2n+1)/(2n+1)! has 2a(n)+1 real roots.
%C A012265 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 A012265 _James Propp_, posting to math-fun mailing list May 30 1997.
%H A012265 Vincenzo Librandi, <a href="/A012265/b012265.txt">Table of n, a(n) for n = 0..200</a>
%t A012265 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 A012265 Cf. A012264.
%K A012265 nonn,easy,nice
%O A012265 0,5
%A A012265 _N. J. A. Sloane_
%E A012265 More terms from _James Sellers_