

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.


3



0, 1, 2, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6, 7, 7
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


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


LINKS

Table of n, a(n) for n=0..14.


EXAMPLE

a(1) = 1 from 1x.
a(2) = 2 from 1+xx^2.
a(3) = 3 from 1xx^2+x^3 = (1x)*(1x^2).
a(5) = 3 from x^5x^4+x^3x^2x+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.
Sequence in context: A097087 A153161 A238516 * A269371 A287355 A194171
Adjacent sequences: A282689 A282690 A282691 * A282693 A282694 A282695


KEYWORD

nonn,more


AUTHOR

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


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


STATUS

approved



