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 A282691 a(n) = maximal number of real roots of any of the 2^(n+1) polynomials c_0 + c_1*x + c_2*x^2 + ... + c_n*x^n where the coefficients c_i are +1 or -1. 2
 0, 1, 2, 3, 2, 3, 2, 5, 4, 5, 4, 5, 4, 5, 6, 7, 6, 5, 6, 7, 6, 7, 6, 9 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The roots are counted with multiplicity. Since (+/-)P((+/-)x) has the same number of real roots as P(x), we need consider only the cases where the x^0 and x^1 coefficients are +1. - Robert Israel, Feb 23 2017 LINKS 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 MAPLE # Maple program using Robert Israel's suggestion (above) for the computation of a(n) using Sturm's Theorem and the squarefree factorization of the 1, -1 polynomials, from W. Edwin Clark, Feb 23 2017: numroots:=proc(p, x) local s: sturm(sturmseq(p, x), x, -infinity, infinity): end proc: a:=proc(n) local m, T, L, L1, p, P, s, k, q, u; m:=0; T:=combinat:-cartprod([seq([1, -1], i=1..n-1)]): while not T[finished] do   L:=T[nextvalue]();   L1:=[1, 1, op(L)];   p:=add(L1[i]*x^(i-1), i=1..n+1);   q:=sqrfree(p, x);   k:=0;   for u in q[2] do k:=k+numroots(u[1], x)*u[2]; od;   if k > m then m:=k;  fi; end do: return m; end proc: MATHEMATICA Do[Print[Max[CountRoots[Internal`FromCoefficientList[#, x], x] & /@  Tuples[{1, -1}, n]]], {n, 1, 23}] (* Luca Petrone, Feb 23 2017 *) CROSSREFS Cf. A282692, A282701. Sequence in context: A064652 A077600 A120223 * A346744 A265577 A251103 Adjacent sequences:  A282688 A282689 A282690 * A282692 A282693 A282694 KEYWORD nonn,more AUTHOR Oanh Nguyen and N. J. A. Sloane, Feb 23 2017 EXTENSIONS a(7) corrected and a(15)-a(16) added by Chai Wah Wu, Feb 23 2017; a(7) also corrected by W. Edwin Clark, Feb 23 2017 a(17)-a(22) added by Luca Petrone, Feb 23 2017 a(23) added by W. Edwin Clark, Feb 24 2017 STATUS approved

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Last modified May 22 09:12 EDT 2022. Contains 353941 sequences. (Running on oeis4.)