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A282692
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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.
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3
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0, 1, 2, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8
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OFFSET
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0,3
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COMMENTS
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The roots are counted with multiplicity.
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
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LINKS
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FORMULA
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EXAMPLE
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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(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
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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