[seqfan] Re: Remarks on lists and offsets

[Fred Lunnon]

<< ... I cannot rule out that it will have precisely the opposite effect >>

    Decisive action in the hour of need
    Denotes the hero, but does not succeed.

                                       (Hilaire Belloc "B stands for
Bear" ; see
    http://www.gutenberg.org/files/40134/40134-h/40134-h.htm

I hereby nominate P. Luschny for the OEIS grizzly (grisly?) medallion,
awarded in recognition of overwhelming determination in the face of
indeterminate residues modulo two.

WFL



On 1/26/21, Peter Luschny <peter.luschny at gmail.com> wrote:
> The recently asked question from John shows some of the
> confusion that keeps cropping up in the context of the choice
> of the offset. The wiki entry [1] is the official description
> of the term 'offset' in the context of the OEIS. Here I would
> like to give a more personal view. It might help to understand
> this quantity better but I cannot rule out that it will have
> precisely the opposite effect.
>
> ---
>
>     The setup:
>     Data:      a,     b,     c,     d,   ...
>     Indices: (0+o), (1+o), (2+o), (3+o), ...
>     where o denotes the offset.
>
> How to choose the right offset? As a first approach, ask
> the question:
>
>    Is the sequence described by a simple formula (function)
>    or by some description (predicate)?
>
> * Case formula/function:
>
> The offset is determined by taking the minimum of the formula's
> maximal range of validity restricted to nonnegative integers.
> Corner cases should be given special attention. The validity
> of the argument '0' for the function should be checked and
> added to the range if valid. The convention that an empty sum
> is 0 should be taken into account.
>
> * Case description/predicate:
>
> The offset is 1. By OEIS convention. The sequence is then
> called a list. What often confuses people in this case is
> that they believe that the predicate's argument range is
> {1,2, ...}. This is wrong. There is nothing implied about
> the range of the predicate by the offset in this case.
>
> ---
>
> A typical case which induces some kind of confusion is when
> the offset = 1 and the first term = 0. Let's trace this more
> formally (with Maple). The predicate 'is_a' determines if 'n'
> is an element of 'a' and returns accordingly 'true' or 'false':
>
>     a := [seq(`if`(is_a(n), n, NULL), n = 0..100)];
>     a[1]; a[2]; a[3]; a[4]; ...
>
> This feels like a kind of cognitive dissonance. The evaluation
> of the predicate starts at n = 0 but the list starts with a[1]!
> But this magic works -- in Maple! Because Maple's lists are
> 1-based. It does not necessarily work in other computer languages.
> For instance a list in Sage/Python is 0-based and here you
> have to work around this difference if you really wish to
> comply with the OEIS convention.
>
> (I have long suspected that the list-offset=1 convention in
> the OEIS might be influenced by Neil's preferred use of Maple.)
>
> ---
>
> Don't expect the above hints to be without exceptions. There is
> this famous sequence where the inventor of the taxonomy writes:
> "Although this is a list, and lists normally have offset 1,
> it seems better to make an exception in this case."
>
> More general, a sequence can have both a simple formula and
> be described by a simple predicate. In such cases, expect
> eternal strife and discord. I know sequences where the offset
> has been changed many times between 0 and 1 for this reason.
> Which in consequence introduced errors in the related formulas.
>
> ---
>
> Here is an example that I recently discussed with Michel
> Dekking [2] in the editor's room:
>
> The predicate is: 'k is a nonnegative integer with an even
> number of 1's in the binary expansion.'
>
> Maple: is_a := n -> add(k, k in convert(n, base, 2))::even:
>
> Two of the following sequences have offset 0 and two have
> offset 1. Homework: Explain which ones and why. [4]
>
> seq(`if`(is_a(n), n, NULL), n=0..20); 0, 3, 5, 6, 9, 10, 12,  ...
> seq(`if`(is_a(n), NULL, n), n=0..20); 1, 2, 4, 7, 8, 11, 13,  ...
> seq(`if`(is_a(n), 1,    0), n=0..20); 1, 0, 0, 1, 0, 1, 1, 0, ...
> seq(`if`(is_a(n), 0,    1), n=0..20); 0, 1, 1, 0, 1, 0, 0, 1, ...
>
> Bonus points: Add the first two sequences termwise. Which
> offset has this compound sequence? The same as the individual
> ones? And if not, is this justified?
>
> ---
>
> One might say that you don't have to like conventions, it's
> enough to stick to them. But they make little sense if they
> can't be followed consistently. How does the list convention
> interact with mathematical operations? Can you add lists
> (termwise) like you can you add sequences? Or is this like
> counting apples and pears together? Assuming the result makes
> sense (as in the example above), is it again a list, or (as
> in the example above) rather a sequence?
>
>     << We see with concern: the curtain is falling
>           and all questions remain unanswered ... >>
>
> Cheers, Peter
>
> [1] https://oeis.org/wiki/Offsets
> [2] https://cs.uwaterloo.ca/journals/JIS/VOL19/Dekking/dekk4.html
> [3] https://arxiv.org/abs/1405.6214
> [4]
> https://oeis.org/search?q=id:A000069|id:A001969|id:A010059|id:A010060|id:A016813&fmt=short
>
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