A282691 a(n) = maximal number of real roots of any of the 2^(n+1) polynomials c_0 + c_1*x + c_2*x^2 + ... + c_n*x^n where the coefficients c_i are +1 or -1.

0, 1, 2, 3, 2, 3, 2, 5, 4, 5, 4, 5, 4, 5, 6, 7, 6, 5, 6, 7, 6, 7, 6, 9, 6, 7, 8, 9, 8, 9, 8

Offset: 0

Author

Oanh Nguyen and N. J. A. Sloane, Feb 23 2017

Keywords

• nonn
• more

Comments

The roots are counted with multiplicity.

Since (+/-)P((+/-)x) has the same number of real roots as P(x), we need consider only the cases where the x^0 and x^1 coefficients are +1. - Robert Israel, Feb 23 2017

Examples

			a(1) = 1 from 1 + x.
a(2) = 2 from 1 + x - x^2.
a(3) = 3 from 1 + x - x^2 - x^3 = (1+x)*(1-x^2).
a(5) = 3 from x^5 - x^4 + x^3 - x^2 - x + 1. - _Robert Israel_, Feb 26 2017
a(7) = 5 from x^7 + x^6 - x^5 - x^4 - x^3 - x^2 + x + 1 = (x - 1)^2*(x + 1)^3*(x^2 + 1). - _Chai Wah Wu_ and _W. Edwin Clark_, Feb 23 2017
		

Crossrefs

Cf. A282692, A282701.

Programs

Maple

# Maple program using Robert Israel's suggestion (above) for the computation of a(n) using Sturm's Theorem and the squarefree factorization of the 1,-1 polynomials, from W. Edwin Clark, Feb 23 2017:
numroots:=proc(p,x)
local s:
sturm(sturmseq(p,x),x,-infinity,infinity):
end proc:
a:=proc(n)
local m,T,L,L1,p,P,s,k,q,u;
m:=0;
T:=combinat:-cartprod([seq([1,-1],i=1..n-1)]):
while not T[finished] do
  L:=T[nextvalue]();
  L1:=[1,1,op(L)];
  p:=add(L1[i]*x^(i-1),i=1..n+1);
  q:=sqrfree(p,x);
  k:=0;
  for u in q[2] do k:=k+numroots(u[1],x)*u[2]; od;
  if k > m then m:=k;  fi;
end do:
return m;
end proc:

Mathematica

Do[Print[Max[CountRoots[Internal`FromCoefficientList[#, x], x] & /@  Tuples[{1, -1}, n]]], {n, 1, 23}] (* Luca Petrone, Feb 23 2017 *)

Sage

import itertools
def a(n):
    if n==0: return 0
    R = ZZ['x']; ans = 0
    for c in itertools.product([-1,1], repeat=(n-1)):
        p = R([1,1]+list(c))
        num_roots = sum([f[0].number_of_real_roots()*f[1] for f in p.factor()])
        ans = max(ans, num_roots)
    return ans  # Robin Visser, Sep 02 2024

Extensions

a(7) corrected and a(15)-a(16) added by Chai Wah Wu, Feb 23 2017; a(7) also corrected by W. Edwin Clark, Feb 23 2017

a(17)-a(22) added by Luca Petrone, Feb 23 2017

a(23) added by W. Edwin Clark, Feb 24 2017

a(24)-a(30) from Robin Visser, Sep 02 2024