[seqfan] Some questions about the %O directive

Sidney Cadot sidney at jigsaw.nl
Thu Sep 24 08:57:08 CEST 2015

Hi all,

I'm in the process of fixing a bunch of minor issues that can be
detected programatically.

I am now starting to check if I can find/fix issues with the %O directive.

For some entries, the "%O" is missing altogether. As far as I
understand this should happen if-and-only-if the sequence has the
keyword 'allocated'; is that correct?

For other entries, the %O line looks like one of these:

%O a,b                (mostly) - or -
%O a                   (only sometimes)

In both cases, the 'a' entry denotes the smallest integer index for
which the sequence is defined, i.e., a(p) is defined only if p >= a.

The second number (if present) indicates the _position_ of the first
sequence value whose absolute magnitude exceeds 1. This always counts
from 1 for the first element in the sequence. So if 'q' denotes the
smallest valid index for which |a(q)|>1, b = q - a + 1. Is this
interpretation correct?

Now as to the presence of the 'b': Charles Greathouse wrote last week:

    "Sequences for which all terms are in {-1, 0, 1} should have only the first
offset number."

However, this is contradictory to the rule found in

    "In the internal format, there is a second offset, which says
which term (counting from the left, and starting with 1), first
exceeds 1 in absolute value. This is set to 1 if all the terms are 0
or +-1. "

This latter rule is reinforced by the explanation given in

    "On the other hand, in this sequence (A010051) no term exceeds 1,
so b takes its default value of 1."

Which rule is the correct one? (I personally feel that the rule given
by Charles is the cleaner one.)

Lastly, there are a few questions on corner cases:

- What do we do if |a(q)| > 1 only for a known large number of q
(beyond the values recorded in the OEIS?)
- What do we do if it is known that |a(q)| > 1 for some value of q,
but the value is not known (or not representable using normal number
notation, e.g. q = 10^(10^100))?
- What do we do if it is unknown whether |a(q)|>1 for any q?

For most sequences, such issues will not occur, but I am curious how
close we can get to a mathematically precise definition of the
intended meaning.

Regards Sidney

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