OFFSET
0,3
COMMENTS
The n-th Bernoulli polynomial is defined by the exponential generating function: t*exp(x*t)/(exp(t)-1) = Sum_{n>=0} bernoulli(n,x)/n!*t^n.
The first few Bernoulli polynomials are:
B(0,x) = 1
B(1,x) = x - 1/2
B(2,x) = x^2 - x + 1/6
B(3,x) = x^3 - 3/2 x^2 + 1/2 x
B(4,x) = x^4 - 2x^3 + x^2 - 1/30
Conjecture 1: for n > 32, a(n) = 3 for n odd and a(n) = 4 otherwise.
Conjecture 2: A284849(n) - a(n) > 0 if n = 6, 7,...,15 and n = 17.
Conjecture 1 is false. It appears that for n => 13, a(n) = 3 for n == 1 (mod 4) and a(n) = 4 otherwise. - Robert Israel, May 29 2017
LINKS
A. P. Veselov and J. P. Ward, On the real zeros of the Hurwitz zeta-function and Bernoulli polynomials, Journal of Mathematical Analysis and Applications 305 (2005), 712-721.
Eric Weisstein's World of Mathematics, Bernoulli Polynomial
FORMULA
Conjectures from Colin Barker, Jan 22 2020: (Start)
G.f.: x*(1 + 2*x + 2*x^2 + 3*x^3 + 2*x^4 + 2*x^9 + 2*x^10 + 2*x^15 - x^19) / ((1 - x)*(1 + x)*(1 + x^2)).
a(n) = a(n-4) for n>16.
(End)
EXAMPLE
a(6) = 2 because the zeros of B(6,x) = x^6 - 3x^5 + 5/2 x^4 - 1/2 x^2 + 1/42 are:
x1 = -0.2728865...-0.06497293...*i,
x2 = -0.2728865...+0.06497293...*i,
x3 = 0.2475407...,
x4 = 0.7524592...,
x5 = 1.272886...-.06497293...*i,
x6 = 1.272886...+.06497293...*i
with two roots x3 and x4 in the open interval (-1, +1).
MAPLE
f:= proc(n) sturm(sturmseq(bernoulli(n, x), x), x, -1, 1) - `if`(n::odd and n > 1, 1, 0) end proc:
map(f, [$0..100]); # Robert Israel, May 29 2017
MATHEMATICA
a[n_] := NSolve[-1 < x < 1 && BernoulliB[n, x] == 0, x, Reals, WorkingPrecision -> 100] // Length;
a /@ Range[0, 100] (* Jean-François Alcover, Oct 26 2020 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, May 22 2017
EXTENSIONS
Corrected by Robert Israel, May 29 2017
STATUS
approved