[seqfan] Re: When is a sequence a list?

[M. F. Hasler]

On Sat, Jul 24, 2021, 08:06 Felix Fröhlich <felix.froe at gmail.com> wrote:

> suppose the sequence b(n) = A001220(n)-1/A305184(n) were in the OEIS. It
> is clear A001220 could be defined by the formula a(n) = A305184(n)*b(n) + 1.
> Wouldn't that mean A001220 would no longer be a list?
>

A set or list becomes or induces a function of n the very moment where you
say "the n-th element of the list".
But what may distinguish a list from a "genuine" function is that here, n
is just an index or label, not really a value you are doing arithmetics
with.
(Although one might use 2n to pick every other element (especially if they
are, e.g., of alternating parity) ; n+3 to omit the first 3 members, etc.
This is because the nonnegative or positive integers we use as indices
because they are the "counting numbers",
and the arithmetics defined on them, have their origin in "physical"
operations on objects,
starting with the "successor" operation [which originally is an operation
on objects [sheep, coins, ...], i.e., elements of a set or list,
but became "arithmeticalized" by considering k-fold repetition of the
operations (so "successor" becomes +k ; then "times k", etc...) ])

Formulas where n is an index/label shouldn't change when all relevant lists
would use any arbitrary different offset (e.g., all start at 0 (or 10)
instead of 1 or vice versa).
This is usually the most efficient criterion to know whether a sequence is
really a function of its argument, or whether that argument is rather an
index/label.


> Does the property of a sequence being a list or not only depend on the
> definition used for the sequence?


Mathematically, a list is a function (assigning a value to each index).
[But, as an quite irrelevant and potentially confusing side note, in the
usual axiomatic approach, functions (like everything else) are actually
sets of sets of sets ... since pairs (Cartesian product) are defined as
sets, (x,y) = { {x}, {x,y} } (Kuratowski), etc...)

So I think what you mean is something like an "arithmetical function",
where the value (result) is obtained by doing arithmetics with the
argument, not just using it as index/label.
But you'll easily see that this is rather philosophical than mathematical,
because any operation like addition, multiplication etc can also be defined
as "table lookup".

So being a list is not a property intrinsic to a sequence? In particular,
> if A001220 is defined in the current way, it is a list, but if it is
> defined as a(n) = A305184(n)*b(n) + 1, it is not?


no, in this simple example the two are not really different, it just means
that the n-th element of the first list is related to the n-th element of
the second and of the third list. Arithmetics is only done on values, but
not on indices.

But as Rainer replied (I was slower...), often our goal is to find an
"explicit expression" of the n-th element of a list / set, and we often
succeed
(as nicely illustrated by the simple example of the list of squares, or
list of odd numbers, etc).
Now THEN there is a "true" ("arithmetic") function of n that gives the n-th
element of the list or set. [Sorry if that may seem contradictory to what I
wrote earlier.]
And we can (and do) identify the function with the list / set.
And for a mathematician, things that are "identified" are the same, there
are just different ways of looking at and writing them.


> I hope someone can help clear this up for me.
>

Summarizing, I'd say that the distinction is rather philosophical,
arbitrary and actually mathematically not very important (not to say
"meaningless").

-Maximilian

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