OFFSET
1,2
COMMENTS
(0,1) (or Newman) polynomials have coefficients from {0,1}.
LINKS
P. Borwein, T. Erdélyi, and G. Kós, Littlewood-type problems on [0,1], Proc. London Math. Soc. 79 (1999), 22-46.
MathOverflow, Number of real roots of 0,1 polynomial.
A. Odlyzko and B. Poonen, Zeros of polynomials with 0,1 coefficients, L’Enseignement Mathématique 39 (1993), 317-348.
FORMULA
a(n) ~ C*n^2 (see Odlyzko and Poonen) with numerical estimate 0.7 < C < 0.9.
MATHEMATICA
(* Suitable only for 1 <= n <= 4;
for larger n, special Julia and Python packages are needed *)
Table[Exponent[Monitor[Catch[Do[
poly = FromDigits[IntegerDigits[k, 2], x];
res = Length@{ToRules@Reduce[poly == 0, x, Reals]};
If[res == n, Throw@{res, Expand@poly}]
, {k, 2000}]], k][[2]], x], {n, 1, 4}]
PROG
(Python)
from itertools import count
from sympy.abc import x
from sympy import sturm, oo, sign, nan, LT
def A368824(n):
for m in count(2):
l = len(s:=bin(m)[2:])
q = sturm(sum(int(s[i])*x**(l-i-1) for i in range(l)))
a = [1 if (k:=LT(p).subs(x, -oo))==nan else sign(k) for p in q[:-1]]+[sign(q[-1])]
b = [1 if (k:=LT(p).subs(x, oo))==nan else sign(k) for p in q[:-1]]+[sign(q[-1])]
if n==sum(1 for i in range(len(a)-1) if a[i]!=a[i+1])-sum(1 for i in range(len(b)-1) if b[i]!=b[i+1]):
return l-1 # Chai Wah Wu, Feb 15 2024
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Denis Ivanov, Jan 07 2024
STATUS
approved