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