Re A094405: John W. Layman's Theorem & Conjection

[Max]

Such sequence stabilizes as soon as
a(1) + a(2) + ... + a(m) = 0 (mod m+2)
for some m. It is easy to see if
a(1) + a(2) + ... + a(m) = k*(m+2)
then a(n) = k for all n>m.

Probabilistic arguments suggest that this will happen sooner or later
for every starting value a(1) but I do not have a proof.

Max

On 2/24/06, zak seidov <zakseidov at yahoo.com> wrote:
> Dear SeqFans,
>
> 1. My conjection:
>
> For any seed a(1) sequence
> "a(n) = (sum of previous terms) mod n"
> ends with repeating constant.
>
> 2. I checked it for a(1) = 1,3,5,..,941.
>
> 2a. Seq's with a(1) = 2m-1 & a(1) = 2m merge
> starting with a(3) hence no need to consider
> cases of  even a(1)'s.
>
> 3. Some examples:
>
> 3a. Sequences with a(1) =
> {9,33,37,39,43,45,47,49,53,55,59,93,97,99,101}
> all have a(n>1241) = 316.
>
> 3b. Sequences with a(1)=
> {449, 451, 455, 457, 471, 473, 475, 479}
> all have a(n>19207) = 4788 (record values1).
>
> 3c. Sequences with a(1) =
>  {1, 51, 103, 107, 109, 129, 133, 137, 165}
> (* including original A094405 *)
> all have a(n>397) = 97.
>
> Any ideas for proof/check/contra-example?
> Zak
>
> %S A094405
> 1,1,2,0,4,2,3,5,0,8,4,6,10,4,5,7,11,1,17,11,18,10,15,1,21,11,16,26,17,
> %T A094405
> 27,16,24,7,5,1,29,13,17,25,1,33,15,20,30,5,45,33,7,2,42,22,32,52,38,8,
> %U A094405
> 2,47,23,32,50,25,35,55,31,46,10,3,57,29,41,65,41,64,36,53,11,2,62,26
> %N A094405 a(1) = 1; a(n) = (sum of previous terms)
> mod n.
> %C A094405 Theorem. For all values of n>=397, a(n)=97.
> Proof. Let s(n) denote Sum[a(i), i=1..n-1].
> Calculation shows that s(397)=38606=397*97+97. Thus
> a(397)=397*97+97 mod 397=97. Then
> s(398)=s(397)+97=398*97+97, giving a(398)=97. A simple
> inductive argument shows that a(397+k)=97 for all
> integers k>=0. - John W. Layman
> (layman(AT)math.vt.edu), Jun 07 2004
> %e A094405 a(4) = 0 because the previous terms 1, 1, 2
> sum to 4, and 4 mod 4 is 0. a(5) = 4 because the
> previous terms 1, 1, 2, 0 sum to 4 and 4 mod 5 is 4.
> %p A094405 L := [1]; s := 1; p := 2; while (nops(L) <
> 90) do; if 1>0 then; t := s mod p; L := [op(L),t]; s
> := s+t; p := p+1; fi; od; L;
> %Y A094405 Sequence in context: A112824 A001100
> A066910 this_sequence A028609 A107490 A079534
> %Y A094405 Adjacent sequences: A094402 A094403 A094404
> this_sequence A094406 A094407 A094408
> %K A094405 nonn
> %O A094405 1,3
> %A A094405 Chuck Seggelin
> (seqfan(AT)plastereddragon.com), Jun 03 2004
>
>
>
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