cp's OEIS Frontend

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.

%I A282692 #62 Jan 28 2022 20:04:11
%S A282692 0,1,2,3,3,3,4,5,5,5,5,5,6,7,7,7,7,7,8,8,8,8
%N 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.
%C A282692 The roots are counted with multiplicity.
%C A282692 Comments from _Chai Wah Wu_, Feb 23 2017: (Start)
%C A282692 1. a(n+1) >= a(n) since p(x)*x has the same number of nonzero real roots as p(x).
%C A282692 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).
%C A282692 (End)
%C A282692 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
%F A282692 a(n) = max { A282701(k) : k=0..n }. - _Max Alekseyev_, Jan 27 2022
%e A282692 a(1) = 1 from 1-x.
%e A282692 a(2) = 2 from 1+x-x^2.
%e A282692 a(3) = 3 from 1-x-x^2+x^3 = (1-x)*(1-x^2).
%e A282692 a(5) = 3 from x^5-x^4+x^3-x^2-x+1. - _Robert Israel_, Feb 26 2017
%e A282692 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
%e A282692 a(8) = 5 from the same polynomial. - _Chai Wah Wu_, Feb 23 2017
%e A282692 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
%Y A282692 Cf. A282691, A282701.
%K A282692 nonn,more
%O A282692 0,3
%A A282692 Oanh Nguyen and _N. J. A. Sloane_, Feb 23 2017
%E A282692 a(7) corrected by _Chai Wah Wu_ and _W. Edwin Clark_, Feb 23 2017
%E A282692 a(8) corrected by _Chai Wah Wu_, Feb 23 2017
%E A282692 a(13)-a(14) corrected by _Chai Wah Wu_, Feb 24 2017
%E A282692 a(15)-a(21) from _Max Alekseyev_, Jan 28 2022