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.

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

Offset: 0

Author

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

Keywords

• nonn
• more

Comments

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

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(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
		

Crossrefs

Cf. A282691, A282692.

Extensions

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