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a(n) = 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, c_0 != 0, and c_n != 0.
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%I #40 Jan 28 2022 20:05:07

%S 0,1,2,3,2,3,4,5,4,5,4,5,6,7,6,7,6,7,8,7,8,7

%N a(n) = 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, c_0 != 0, and c_n != 0.

%C The roots are counted with multiplicity (and are nonzero, by definition).

%C Unlike A282692, this sequence is not monotonic.

%C A282692(n) >= a(n) >= A282691(n). A282692(n) = max(A282692(n-1),a(n)). Differs from A282691 for n = 6, 12, 13 (and most likely other values of n). - _Chai Wah Wu_, Feb 25 2017

%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(13) = 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, A282692.

%K nonn,more

%O 0,3

%A Oanh Nguyen and _N. J. A. Sloane_, Feb 23 2017

%E a(13) corrected by _Chai Wah Wu_, Feb 25 2017

%E a(15)-a(16) added by _Luca Petrone_, Feb 26 2017

%E a(17)-a(21) from _Max Alekseyev_, Jan 28 2022