login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 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.

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 20 18:31 EST 2018. Contains 299381 sequences. (Running on oeis4.)