# Longest Lucas seq. from start to "n"

Max Alekseyev maxale at gmail.com
Thu Oct 18 12:55:31 CEST 2007

```Still, if it is a Lucas sequence then the initial term must be 0 (ifit is U_n(P,Q)) or 2 (if it is V_n(P,Q)). Moreover, in all yourexamples the terms of a sequence satisfy Fibonacci-like recurrence:u(n+2)=u(n)+u(n+1).So, do you want a longest linear recurrent sequence withu(n+2)=u(n)+u(n+1) with all terms non-negative and the last term equalto the given number? In general case, it won't be a Lucas sequence.
Max
On 10/18/07, Eric Angelini <Eric.Angelini at kntv.be> wrote:>> Yes, sorry Max -- no Lucas numbers, no negative terms.>> 15 1 16 17> 13 2 15 17> 11 3 14 17>  9 4 13 17>  7 5 12 17>  4 1  5  6 11 17 <-- "Llatest entry of integer n=17 in a Lucas seq">  4 3  7 10 17>  7 1  8  9 17>> Best,> É.>>>> -----Message d'origine-----> De : Max Alekseyev [mailto:maxale at gmail.com]> Envoyé : jeudi 18 octobre 2007 12:39> À : Eric Angelini> Cc : seqfan at ext.jussieu.fr> Objet : Re: Longest Lucas seq. from start to "n">> Eric,>> Your problem formulation is unclear.>> Do you wand a sequence of Lucas numbers or (generic) Lucas sequence?> These are different beasts:> http://mathworld.wolfram.com/LucasNumber.html> http://mathworld.wolfram.com/LucasSequence.html>> Do you want all elements of your sequence be positive (or> non-negative) integers?>> Max>> On 10/18/07, Eric Angelini <Eric.Angelini at kntv.be> wrote:> >> > Hello SeqFans,> > I guess this is (very) old hat -- sorry.> >> > I want to insert "17" (for instance) as later as> > possible in a Lucas seq. I found by hand that the> > seq 4 1 5 6 11 [17] 28...  is the answer (5 terms> > before "17"; the seq 7 1 8 9 [17] 26... has 4)> >> > What is the general method to find those? There> > must be a seq in the OEIS linking n to the length> > of its "latest entry" in a Lucas sequence -- but> > I cannot find it.> >> > What would be the longest such chain including 2007?> >> > Best,> > É.> >> >>>

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