A little fun with Fibonacci, tau, etc.

[benoit]

> Now, I had the idea building the sequence of positions of terms
> differing, in this way:
> "take A007064, consider initial offset is 1, and then look if
> A007064(n) = A057843(n+1) or not; in case they are different, take n
> in the new sequence".
>
> It makes:
> 1 4 6 9 11 12 14 17 19 22 25 27 30 32 33 35 38 40 43 46 48 51

Interesting. That's the sequence of n such that 
floor(phi*n-phi/2)=floor(phi*n+phi-3)

This sequence has an asymptotic behaviour a(n)=c*n with 
c=2.341640786499873817845504198...=1+3/sqrt(5)

Unfortunately there seems not to exist a closed form for a(n) as 
floor(c*n+d)

More generally sequence of n such that floor(phi*n+x)=floor(phi*n+y) 
for suitable x and y  and   y>x satisifies asymptotically 
a(n)=n/(1-y+x)+O(1)

An exemple already in the database A005653 : that's also the sequence 
of k such that floor(phi*k)=floor(phi*k+1/2) and here  
A005653(n)=2*n+O(1)

More generally, given an irrational r, and suitable x and y, y>x we 
have the same asymptotic rule :

sequence (a(n)) of k such that floor(r*k+x)=floor(r*k+y)  satisfies 
a(n)=c*n+O(1) with c=1/(1-y+x)

rmk : my O(1) should be rigorously proven.

BC.





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