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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, 7, 7, 7, 8, 8, 8, 8 (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
FORMULA
a(n) = max { A282701(k) : k=0..n }. - Max Alekseyev, Jan 27 2022
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
Sequence in context: A153161 A346373 A238516 * A269371 A287355 A194171
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
a(15)-a(21) from Max Alekseyev, Jan 28 2022
STATUS
approved

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Last modified December 7 07:48 EST 2023. Contains 367630 sequences. (Running on oeis4.)