A282692 a(n) = maximal number of nonzero real roots of any of the 3^(n+1) polynomials c_0 + c_1*x + c_2*x^2 + ... + c_n*x^n where the coefficients c_i are -1, 0, or 1.

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

Offset: 0

Author

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

Keywords

• nonn
• more

Comments

The roots are counted with multiplicity.

Comments from Chai Wah Wu, Feb 23 2017: (Start)

1. a(n+1) >= a(n) since p(x)*x has the same number of nonzero real roots as p(x).

2. If we define a sequence b(n) by requiring the highest coefficient to be nonzero, that is, if we let b(n) = maximal number of nonzero 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, and c_n != 0, then Comment 1 shows that we get nothing new, and b(n) = a(n).

(End)

From the reasoning in Chai Wah Wu's comment 1, this is also the 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, and c_0 != 0. A new sequence b(n) is created (A282701) if both c_0 and c_n are != 0. - Peter Munn, Feb 25 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
a(8) = 5 from the same polynomial. - _Chai Wah Wu_, Feb 23 2017
a(13) = a(14) = 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
		

Crossrefs

Cf. A282691, A282701.

Formula

a(n) = max { A282701(k) : k=0..n }. - Max Alekseyev, Jan 27 2022

Extensions

a(7) corrected by Chai Wah Wu and W. Edwin Clark, Feb 23 2017

a(8) corrected by Chai Wah Wu, Feb 23 2017

a(13)-a(14) corrected by Chai Wah Wu, Feb 24 2017

a(15)-a(21) from Max Alekseyev, Jan 28 2022