[seqfan] Re: name for class of sequences
Frank Adams-Watters
franktaw at netscape.net
Wed Jan 21 22:33:01 CET 2015
I don't know about a name, but these are precisely the sequences whose
generating function is the reciprocal of a monic polynomial.
Franklin T. Adams-Watters
-----Original Message-----
From: Bob Selcoe <rselcoe at entouchonline.net>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Wed, Jan 21, 2015 3:28 pm
Subject: [seqfan] name for class of sequences
Hello Seqfans,
Consider a class of sequences generated by the recurrence:
a(n) = q*a(n-f1) + r*a(n-f2) + s*a(n-f3)... + y*a(n-fz)
where:
i. the only seed value is a(0) = 1, and a(n) n < 0 = 0;
ii. f1..fz and q,r,s,..y are positive integers, and
iii. f1 < f2 .. < fz
Among the most basic sequences in this class are:
A000045 Fibonacci: a(n) = a(n-1) + a(n-2);
A000931 Padovan: a(n) = a(n-2) + a(n-3); and
A000129 Pell numbers: a(n) = 2*a(n-1) + a(n-2)
The class could be called "ordered partition sequences" or "composition
sequences" because for all such sequences, a(n) is the number of
compositions (ordered partitions) of n into q sorts of f1's, r sorts
of
f2's, s sorts of f3's... y sorts of fz's.
This may be very basic, but does a name for such a class already exist?
Thanks,
Bob Selcoe
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