# [seqfan] Offsets for Recurrence Sequences

Creighton Kenneth Dement creighton.k.dement at mail.uni-oldenburg.de
Thu Apr 23 02:08:40 CEST 2009

```Dear Seqfans,

I apologize if this is a silly idea or has been discussed before, but I
would like to propose that from now on every newly submitted sequence
which is a linear recurrence is given a 0 offset and those which can be
changed without messing up existing formula (or combinatorial definitions)
sequences are linked by an equation, it is more time consuming to check
when offsets differ. In the past, when I think a reader may become
confused, I explicitely write the offset: A_0(n+1) + C_1(n+3) = B_0(n) -
but then one presumably needs at least one more sentence to explain the
notation.

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

Here is my own concrete example from today.

A098301, offset 0 : [1, 16, 225, 3136, 43681, 608400, 8473921, 118026496,
1643897025, 22896531856]
Member r=16 of the family of Chebyshev sequences S_r(n) defined in A092184.

A011916, offset 1: [3, 44, 615, 8568, 119339, 1662180, 23151183, 322454384]
Integers n such that n^2 = sum(n+1,n+2,n+3,...,n+x) for some value of x.

A123480, offset 1: [4, 60, 840, 11704, 163020, 2270580, 31625104, 440480880]
Coefficients of the series giving the best rational approximations to sqrt(3)

A quick glance reveals A098301 + A011916 = A123480 and what should be a
simple case of sending in a comment becomes more difficult because the
offsets are different. Would it be o.k. just to write A098301 + A011916 =
A123480, offsets differ?

Sincerely,
Creighton

```