[seqfan] Re: Offsets for Recurrence Sequences

[Richard Mathar]

ckd> From seqfan-bounces at list.seqfan.eu Thu Apr 23 02:27:24 2009
ckd> Date: Thu, 23 Apr 2009 02:08:40 +0200 (CEST)
ckd> From: "Creighton Kenneth Dement" <creighton.k.dement at mail.uni-oldenburg.de>
ckd> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
ckd> Subject: [seqfan]  Offsets for Recurrence Sequences
ckd> ...
ckd> I apologize if this is a silly idea or has been discussed before, but I
ckd> would like to propose that from now on every newly submitted sequence
ckd> which is a linear recurrence is given a 0 offset and those which can be
ckd> changed without messing up existing formula (or combinatorial definitions)
ckd> are changed.

This is not a good idea, because in many cases the definitions
start to make sense only if the offset is 1. If we would start all of them
with 0, in a good deal of cases one would have to introduce an artificial,
almost randomly chosen value for a(0), which would complicate look-up,
obscure duplicates etc. The need to explain what that particular a(0) is
chosen in some hand-waving comment is to be avoided. We have had discussions
of this kind in the past, typically involving questions like counting
elements in empty sets, prime signatures of constants like 0 and 1 etc.

ckd> For example, if I want to leave a comment that 3 or more
ckd> sequences are linked by an equation, it is more time consuming to check
ckd> when offsets differ. In the past, when I think a reader may become
ckd> confused, I explicitely write the offset: A_0(n+1) + C_1(n+3) = B_0(n) -
ckd> but then one presumably needs at least one more sentence to explain the
ckd> notation.

There is no need for this; the reader would not become confused if the
proper notation, including correctly type-set indices, were chosen.

ckd> Why are there two essentially the same (core) Lucas sequences with offsets
ckd> 0 (A000032) and 1 (A000204) but apparently no corresponding Fibonacci
ckd> sequence with an offset of 1? For example, A000032 has a comment "Starting
ckd> (1, 3, 4, 7, 11,...) = row sums of triangle A131774." with no mention of
ckd> this for A000204. IMHO, it seems needlessly confusing.

This is confusing because the comment is of poor quality. Of course, the
comment should say "row sums of row n are A000032(n)", aligning the
(correct) notation of the row-index in the table/triangle
with the (correct) index in the linear sequence.

ckd> Here is my own concrete example from today.
ckd> 
ckd> A098301, offset 0 : [1, 16, 225, 3136, 43681, 608400, 8473921, 118026496,
ckd> 1643897025, 22896531856]
ckd> Member r=16 of the family of Chebyshev sequences S_r(n) defined in A092184.
ckd> 
ckd> A011916, offset 1: [3, 44, 615, 8568, 119339, 1662180, 23151183, 322454384]
ckd> Integers n such that n^2 = sum(n+1,n+2,n+3,...,n+x) for some value of x.
ckd> 
ckd> A123480, offset 1: [4, 60, 840, 11704, 163020, 2270580, 31625104, 440480880]
ckd> Coefficients of the series giving the best rational approximations to sqrt(3)
ckd> 
ckd> 
ckd> A quick glance reveals A098301 + A011916 = A123480 and what should be a
ckd> simple case of sending in a comment becomes more difficult because the
ckd> offsets are different. Would it be o.k. just to write A098301 + A011916 =
ckd> A123480, offsets differ?

This just reveals some laziness. The notation without indices should be left
to constants (which one can add,multiply, subtract etc) and perhaps to
union, intersection etc of sequences, but the explicit words like UNION,
SET-MINUS, INTERSECT or \ are clearer if sequences are mixed/complemented etc.
Note that indices, in the mathematical context and in any programming
languages, DO matter. Leaving this information out does not make sense and
just pushes the burden to get them right to the OEIS user (or editors).
It is needed by any reader who has to deal with the sequences, anyway.

RJM