login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Table of n, a(n) for n=1..10.

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

Cf. A206074, A206443.

Sequence in context: A139974 A142188 A248409 * A156156 A201486 A176617

Adjacent sequences:  A206441 A206442 A206443 * A206445 A206446 A206447

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 07 2012

EXTENSIONS

a(8)-a(10) from Robert G. Wilson v, Feb 11 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 3 16:28 EDT 2021. Contains 346439 sequences. (Running on oeis4.)