[seqfan] Re: A1595 mentioned in Medium article
David Seal
david.j.seal at gwynmop.com
Mon Nov 9 10:46:59 CET 2020
> On 09/11/2020 05:34 Alonso Del Arte <alonso.delarte at gmail.com> wrote:
> ...
> I've been wondering about Fibonacci(*n*) − A001595(*n*). That's probably
> already in the OEIS, though maybe without signs. It does change sign, right?
>From the definition of the Fibonacci numbers (A000045), which is "F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1", and from the definition of A001595, which is "a(n) = a(n-1) + a(n-2) + 1, with a(0) = a(1) = 1", it is very easy to prove by induction that a(n) >= F(n) for all n >= 0 (with equality if and only if n = 1), and so the difference of the two sequences does not change sign.
Is there some subtlety about this question whose significance I'm missing? - for instance, a meaning of "*n*"?
A search for the first ten values of A001595(n) - A000045(n), which are 1,0,2,3,6,10,17,28,46,75, says that it doesn't match any sequence in the OEIS. However, leaving off its initial 1 finds A001610, described as "a(n) = a(n-1) + a(n-2) + 1" - an incomplete definition, so I have submitted a change to add "with a(0) = 0 and a(1) = 2". With that completed definition, it's also very easy to prove by induction that A001610(n) = A001595(n+1) - A000045(n+1) for all n >= 0.
David
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