a(m+1)=sum{k|m}mu(k)a(k), & Hidden Negatives

[Leroy Quet]

Alec Mihailovs wrote:
>> From: Leroy Quet [mailto:qq-quet at mindspring.com]
>> Sent: Tuesday, August 17, 2004 3:04 PM
> 
>> Let a(1) = 1;
>> 
>> Let a(m+1) = sum{k|m} mu(k) a(k),
>> 
>> where mu is the Mobius (Moebius) function.
>> 
>> I get, by hand, the sequence beginning:
>> 
>> 1,1,0,1,0,1,1,0,0,1,1,1,1,1,0,1,0,...
>
>The 12th term is wrong. It should be a(12)=a(1)-a(11)=1-1=0. Further,
>a(14)=a(1)-a(13)=1-1=0, and a(15)=a(1)-a(2)-a(7)+a(14)=1-1-1+0=-1, then
>a(16)=a(1)-a(3)-a(5)+a(15)=1-0-0+(-1)=0. Here are the first 100 terms
>calculated in Maple, 
>
>a:=proc(n) option remember;
>add(numtheory[mobius](i)*a(i),i in numtheory[divisors](n-1)) end:
>a(1):=1:
>seq(a(n),n=1..100);
>
>1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, -1, 0, 0, 1, 1, 0, 1, 1, 0, 1,
>1, 1, 0, 1, -1, 2, -1, 2, 0, 0, 0, 0, 1, 0, -1, -1, 1, 0, 1, 0, 0, 0, 0,
>1, 1, 0, 1, 2, 0, 1, 1, 1, -1, -1, 0, 1, -1, 2, 3, 1, 0, 0, 1, 0, 0, 1,
>-1, 2, 1, 0, -1, 0, -1, -2, 2, -1, 1, 1, 0, 1, 1, 2, 1, 3, 0, 1, -1, -2,
>0, 2, 2, 2, 1, 0, -1, 0
> 
>> I get that a(52) = 2, so not every term is either 0 or 1.
>> 
>> (I am betting that not every term is necessarily nonnegative either.)
>> 
>> Could someone extend this sequence?
>> When is the first negative term?
>



I *hate* when I miscalculate terms of a sequence.
Sorry about that.
:/

When I enter the REAL first few terms into the OEIS, I get that this 
sequence is already in the database (sequence A070965).
And furthermore, it was *I* who submitted the sequence back in May of 
2002, apparently.
:)

thanks, sorry,
Leroy Quet