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A206444 Least n such that L(n)<-1 and L(n) where L(k) means the least root of the polynomial p(k,x) defined at A206284, and a(1)=13.

%I A206444 #16 Mar 14 2014 02:01:13
%S A206444 13,53,213,853,3413,13653,54613,218453,873813,3495253
%N A206444 Least n such that L(n)<-1 and L(n)<L(n-1), where L(k) means the least root of the polynomial p(k,x) defined at A206284, and a(1)=13.
%C A206444 A206074 gives an ordering {p(n,x)} of the polynomials with coefficients in {0,1}.  The least n for which p(n,x) has a root r less than -1 is 13, hence the choice of 13 as the initial term of A206443.  (Specifically, p(13,x)=1+x^2+x^3, and r=-1.46557...)  The next p(n,x) having a root less than -1 and <r is p(53,x)=1+x^2+x^4+x^5, with least root -1.57014...
%C A206444 The first 10 terms of A206444 are also the 2nd through 11th terms of A072197.
%t A206444 highs := {First /@ #, Most[FoldList[Plus, 1, Length /@ #]]} &[Split[Rest[FoldList[Max, -\[Infinity], #]]]] &
%t A206444 f[polyInX_] := {Min[#], Max[#]} &[
%t A206444   Map[#[[1]] &, DeleteCases[Map[{#, Head[#]} &, Chop[N[x /. Solve[polyInX == 0, x], 40]]], {_, Complex}]]]
%t A206444 t = Table[IntegerDigits[n, 2], {n, 1, 100000}];
%t A206444 b[n_] := Reverse[Array[x^(# - 1) &, {n + 1}]]
%t A206444 p[n_] := t[[n]].b[-1 + Length[t[[n]]]]
%t A206444 Table[p[n], {n, 1, 25}]
%t A206444 fitCriterion = Intersection[Map[#[[1]] &, DeleteCases[
%t A206444        Table[{n, Boole[IrreduciblePolynomialQ[p[n]]]}, {n, 1, #}], {_, 0}]], Map[#[[1]] &, DeleteCases[
%t A206444        Table[{n, CountRoots[#, {x, -Infinity, 0}] -
%t A206444        CountRoots[#, {x, -1, 0}] &[p[n]]}, {n, 1, #}],
%t A206444            {_, 0}]]] &[Length[t]];
%t A206444 polyNum = Map[{f[p[#]][[1]], #} &, fitCriterion];
%t A206444 up = Map[polyNum[[#]] &, highs[Map[#[[1]] &, polyNum]][[2]]]
%t A206444 down = Map[polyNum[[#]] &, highs[Map[#[[1]] &, -polyNum]][[2]]]
%t A206444 Table[up[[k, 2]], {k, 1, Length[up]}]      (* A206443 *)
%t A206444 Table[down[[k, 2]], {k, 1, Length[down]}]  (* A206444 *)
%t A206444 (* _Peter J. C. Moses_, Feb 06 2012 *)
%Y A206444 Cf. A206074, A206443.
%K A206444 nonn
%O A206444 1,1
%A A206444 _Clark Kimberling_, Feb 07 2012
%E A206444 a(8)-a(10) from _Robert G. Wilson v_, Feb 11 2012