[seqfan] Re: A1595 mentioned in Medium article
Neil Sloane
njasloane at gmail.com
Mon Nov 9 17:39:51 CET 2020
David Seal said: "A001610, described as "a(n) = a(n-1) + a(n-2) + 1" - an
incomplete definition," Note that this sequence says "(Formerly M0764
N0291)", which means it was in the 1973 Handbook of Integer Sequences, and
in those days - and later - the convention was that if the initial terms
were not specified then they were understood to be the initial terms that
you could see at the beginning of the data. The definition was not
incomplete!
Neil
On Mon, Nov 9, 2020 at 8:47 AM David Seal <david.j.seal at gwynmop.com> wrote:
> > 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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