A319116 - OEIS

%I #22 Apr 21 2024 22:26:23

%S 1,-1,-1,1,1,-1,-1,1,1,-1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,1,-1,-1,1,1,

%T -1,-1,1,1,-1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,1,-1,-1,1,1,-1,-1,1,1,-1,

%U -1,-1,1,1,-1,-1,1,1,-1,-1,1,1,1,-1,-1,1,1,-1,-1,1,1

%N Signs of the Maclaurin coefficients of 1/(exp(x) + Pi/2).

%F Conjecture: a(n) is the same as the sign of cos((n+1)*arctan(Pi/log(Pi/2))). We can also use arctan(Pi/log(Pi/2))/Pi ~ 1/2 ~ 5/11 ~ 3776/8307 ~ 3781/8318 and so on. If the numerator is odd, we have, for example, a(n) = -a(n+2) or a(n) = -a(n+11) with some counterexamples. For even numerators we have, for example, a(n) = a(n+8307), also with some counterexamples.

%e a(1) = -a(12) = a(23) = -1.

%p S:= series(1/(exp(x)+Pi/2), x, 147):

%p seq(signum(coeff(S, x, j)), j=0..146);

%t Sign[CoefficientList[Series[1/(Exp[x] + Pi/2), {x, 0, 100}], x]] (* _G. C. Greubel_, Oct 31 2018 *)

%o (PARI) apply(x->sign(x), Vec(1/(exp(x+O(x^100)) + Pi/2))) \\ _Michel Marcus_, Sep 13 2018

%Y Cf. A210245, A210247.

%K sign

%O 0

%A _Mikhail Kurkov_, Sep 10 2018 [verification needed]