[seqfan] Re: New sequences from generalized Fibonacci sequence
Kerry Mitchell
lkmitch at gmail.com
Sun Nov 9 20:07:32 CET 2014
Hi Dale,
When I started playing with this idea, I approached it from the perspective
of a signature sequence and took the postive-integer approach. I also only
wanted to deal with real roots, as they reflect that actual ratios seen
with real-valued Fibonacci-type sequences. But certainly, this could be
extended to non-positive b values.
Kerry
On Sun, Nov 9, 2014 at 2:45 AM, Dale Gerdemann <dale.gerdemann at gmail.com>
wrote:
> Hello Kerry, Hello Seqfans,
>
> Can you explain why you want only positive values for b?
>
> Dale
>
>
> On Sunday, November 9, 2014, Kerry Mitchell <lkmitch at gmail.com> wrote:
>
> > Hi all,
> >
> > Here's something I've been playing with lately. Background: the
> Fibonacci
> > sequence is defined by the recurrence f(n+1) = f(n) + f(n-1). As n
> > increases, the ratio of consecutive terms approaches the golden ratio,
> Phi
> > ~ 1.618. This is because, if we assume that f(n+1) = r x f(n), then the
> > recurrence becomes r^2 = r + 1, and Phi is the positive root of that
> > quadratic.
> >
> > Generalizing, let f(n+1) = a f(n) + b f(n-1). Then, the recurrence
> > quadratic becomes r^2 = a r + b. The ratio of consecutive terms
> approaches
> > r = (a + sqrt(a^2 + 4b))/2. To create the sequences, let a and b be
> > positive integers. List a, b, and r and sort by increasing values of r.
> > In the case of a tie, sort by increasing values of sqrt(a^2 + b^2). The
> > first several sorted values of a, b, r, and sqrt(a^2+b^2) are:
> >
> > 1. 1, 1, 1.618, 1.414
> > 2. 1, 2, 2, 2.236
> > 3. 1, 3, 2.303, 3.162
> > 4. 2, 1, 2.414, 2.236
> > 5. 1, 4, 2.562, 4.123
> > 6. 2, 2, 2.732, 2.828
> > 7. 1, 5, 2.791, 5.099
> > 8. 2, 3, 3, 3.606
> > 9. 1, 6, 3, 6.083
> > 10. 1, 7, 3.193, 7.071
> >
> > The a sequence begins: 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 2, 1, 3,
> 2,
> > 1, 3, 2, 1, 3, 2, 1, 1, 2, 3, 4, 1, 2, 3, 1, 4, 2, 1, 3, 2, 4, 1, 3, 2,
> 1,
> > 4, 3, 2, 1, 4, 3, 2, 1, 1, 2.
> >
> > The b sequence begins: 1, 2, 3, 1, 4, 2, 5, 3, 6, 7, 4, 1, 8, 5, 9, 2,
> 6,
> > 10, 3, 7, 11, 4, 8, 12, 13, 9, 5, 1, 14, 10, 6, 15, 2, 11, 16, 7, 12, 3,
> > 17, 8, 13, 18, 4, 9, 14, 19, 5, 10, 15, 20, 21, 16.
> >
> > Neither sequence submits to upper trimming or lower trimming, as fractal
> > sequences do. Each sequence does count the other--the a term is how many
> > times the corresponding b term has occured in the b sequence, and vice
> > versa. The associative arrays of a and b are transposes of each other
> and
> > the first column of the b's associative array, 1, 4, 12, 28,55, 96, etc.,
> > seems to be A006000. The first column of a's associative array is not in
> > the OEIS.
> >
> > Are these sequences interesting enough to submit to OEIS?
> >
> > Thanks,
> > Kerry Mitchell
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
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