[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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