[seqfan] Re: New sequences from generalized Fibonacci sequence

Dale Gerdemann dale.gerdemann at gmail.com
Sun Nov 9 10:45:59 CET 2014


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
>
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