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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. 2
13, 53, 213, 853, 3413, 13653, 54613, 218453, 873813, 3495253 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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...
The first 10 terms of A206444 are also the 2nd through 11th terms of A072197.
LINKS
MATHEMATICA
highs := {First /@ #, Most[FoldList[Plus, 1, Length /@ #]]} &[Split[Rest[FoldList[Max, -\[Infinity], #]]]] &
f[polyInX_] := {Min[#], Max[#]} &[
Map[#[[1]] &, DeleteCases[Map[{#, Head[#]} &, Chop[N[x /. Solve[polyInX == 0, x], 40]]], {_, Complex}]]]
t = Table[IntegerDigits[n, 2], {n, 1, 100000}];
b[n_] := Reverse[Array[x^(# - 1) &, {n + 1}]]
p[n_] := t[[n]].b[-1 + Length[t[[n]]]]
Table[p[n], {n, 1, 25}]
fitCriterion = Intersection[Map[#[[1]] &, DeleteCases[
Table[{n, Boole[IrreduciblePolynomialQ[p[n]]]}, {n, 1, #}], {_, 0}]], Map[#[[1]] &, DeleteCases[
Table[{n, CountRoots[#, {x, -Infinity, 0}] -
CountRoots[#, {x, -1, 0}] &[p[n]]}, {n, 1, #}],
{_, 0}]]] &[Length[t]];
polyNum = Map[{f[p[#]][[1]], #} &, fitCriterion];
up = Map[polyNum[[#]] &, highs[Map[#[[1]] &, polyNum]][[2]]]
down = Map[polyNum[[#]] &, highs[Map[#[[1]] &, -polyNum]][[2]]]
Table[up[[k, 2]], {k, 1, Length[up]}] (* A206443 *)
Table[down[[k, 2]], {k, 1, Length[down]}] (* A206444 *)
(* Peter J. C. Moses, Feb 06 2012 *)
CROSSREFS
Sequence in context: A139974 A142188 A248409 * A156156 A201486 A176617
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 07 2012
EXTENSIONS
a(8)-a(10) from Robert G. Wilson v, Feb 11 2012
STATUS
approved

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Last modified July 21 18:17 EDT 2024. Contains 374475 sequences. (Running on oeis4.)