[seqfan] Re: New sequences from generalized Fibonacci sequence

Frank Adams-Watters franktaw at netscape.net
Sun Nov 9 06:35:31 CET 2014


I've been calling that operation - counting the number of times the 
value has occurred in the source sequence - the ordinal transform. You 
can find a number of occurrences of this in the OEIS.

And yes, I do think that those sequences are worth adding.

Franklin T. Adams-Watters

-----Original Message-----
From: Kerry Mitchell <lkmitch at gmail.com>
To: seqfan <seqfan at list.seqfan.eu>
Sent: Sat, Nov 8, 2014 11:13 pm
Subject: [seqfan] New sequences from generalized Fibonacci sequence


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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