[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)
are changed. For example, if I want to leave a comment that 3 or more
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
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