A282692 -id:A282692 - cp's OEIS frontend

A282691 a(n) = maximal number of real roots of any of the 2^(n+1) polynomials c_0 + c_1x + c_2x^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

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

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

			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

Cf. A282692, A282701.

# 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:

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

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

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

A282701 a(n) = maximal number of real roots of any of the polynomials c_0 + c_1x + c_2x^2 + ... + c_n*x^n where the coefficients c_i are -1, 0, or 1, c_0 != 0, and c_n != 0.

Original entry on oeis.org

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

Offset: 0

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

The roots are counted with multiplicity (and are nonzero, by definition).
Unlike A282692, this sequence is not monotonic.
A282692(n) >= a(n) >= A282691(n). A282692(n) = max(A282692(n-1),a(n)). Differs from A282691 for n = 6, 12, 13 (and most likely other values of n). - Chai Wah Wu, Feb 25 2017

			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
a(13) = 7 from x^13 + x^12 - x^11 - x^10 - x^9 - x^8 + x^5 + x^4 + x^3 + x^2 - x - 1 = (x - 1)^3*(x + 1)^4*(x^2 + 1)*(x^2 - x + 1)*(x^2 + x + 1). - _Chai Wah Wu_, Feb 24 2017

Cf. A282691, A282692.

a(13) corrected by Chai Wah Wu, Feb 25 2017
a(15)-a(16) added by Luca Petrone, Feb 26 2017
a(17)-a(21) from Max Alekseyev, Jan 28 2022

A284384 Alternate A167381(n), A167381(n) + 1.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 6, 7, 10, 11, 14, 15, 18, 19, 23, 24, 29, 30, 35, 36, 41, 42, 47, 48, 53, 54, 60, 61, 68, 69, 76, 77, 84, 85, 92, 93, 100, 101, 108, 109, 116, 117, 125, 126, 135, 136, 145, 146, 155, 156, 165, 166, 175, 176, 185, 186, 195, 196, 205, 206, 215, 216

Offset: 0

Paul Curtz, Mar 26 2017

nonn

The first differences (a(n+1) - a(n)) are b(n) = 1, 0, 1, 1, 1, 2, 1, 3, 1, 3, 1, 3, 1, 4, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 6, 1, 7, 1, 7, ... .
b(2n+1) = 0, 1, 2, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6, 7, 7, 7, 7, 7, 7, 7, 8, ... =
0, 1,
2, 3, 3, 3
4, 5, 5, 5, 5, 5,
6, 7, 7, 7, 7, 7, 7, 7,
...
= 2*n followed by 2*n+1 times 2*n+1 (by rows) = A282692(n)?
= A284359(n) - 1.

Cf. A005408, A167381, A282692, A284359.

cp's OEIS Frontend

A282691 a(n) = maximal number of real roots of any of the 2^(n+1) polynomials c_0 + c_1x + c_2x^2 + ... + c_n*x^n where the coefficients c_i are +1 or -1.

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A282701 a(n) = maximal number of real roots of any of the polynomials c_0 + c_1x + c_2x^2 + ... + c_n*x^n where the coefficients c_i are -1, 0, or 1, c_0 != 0, and c_n != 0.

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A284384 Alternate A167381(n), A167381(n) + 1.

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cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Sage

Extensions

A282701 a(n) = maximal number of real roots of any of the polynomials c_0 + c_1*x + c_2*x^2 + ... + c_n*x^n where the coefficients c_i are -1, 0, or 1, c_0 != 0, and c_n != 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A284384 Alternate A167381(n), A167381(n) + 1.

Views

Author

Keywords

Comments

Crossrefs

A282691 a(n) = maximal number of real roots of any of the 2^(n+1) polynomials c_0 + c_1x + c_2x^2 + ... + c_n*x^n where the coefficients c_i are +1 or -1.

A282701 a(n) = maximal number of real roots of any of the polynomials c_0 + c_1x + c_2x^2 + ... + c_n*x^n where the coefficients c_i are -1, 0, or 1, c_0 != 0, and c_n != 0.