extending ill-defined sequences

[David Wasserman]

Dear fellow Seqfans,
    I've spent a lot of time working on the list of "Sequences that need
extending", and frequently I find entries where the name gives some
properties of the sequence, but these properties aren't enough to
determine the sequence.  Is there a proper way to extend such sequences?
    Here's how I've done it in the past: usually I can find a simple
rule that explains almost all of the choices.  For example, a(n) might
be the smallest positive integer that fits all the requirements, or if
the sequence is strictly increasing, a(n) might be the smallest number >
a(n - 1) that fits all the requirements.  If there are 1 or 2 members
that don't fit the rule, I usually assume they're mistakes, and correct
them.
    There are a few problematic examples like A082006 shown below.  In
this case, every member shown is the smallest positive integer not
already used that works.  However, this "greedy algorithm" eventually
fails: the next 3 terms are 29, 37, 41, and then to finish the 6th
antidiagonal, we need a number that is congruent to 3 mod 6, but coprime
to 3, which is impossible.  So to continue the sequence, we have to
backtrack and change the 41 (perhaps to 43).  I could decide that I am
finding the lexicographically minimal sequence that fits the
constraints, but that seems rather arbitrary, and not necessarily
consistent with the sequence author's intentions.  
    Any suggestions?

 - David

%I A082006
%S A082006 1,2,4,3,5,7,9,11,13,19,17,21,8,23,31,25,27
%N A082006 In the following square array numbers (not occurring earlier)
are entered like this: a(1, 1), a(1, 2), a(2, \
  1), a(3, 1), a(2, 2), a(1, 3), a(1, 4), a(2, 3), a(3, 2), a(4, 1),
a(5, 1), a(4, 2), ... such that every entry is cop\
  rime to the members of the row and column it belongs, with the
condition that the n-th diagonal sum is a multiple of \
  n. 1 2 7 9 31 25... 4 5 11 23 27... 3 13 8... 19 21... 17 ... ...
Sequence contains numbers as they are entered. %Y A082006 Cf. A082007,
A082008, A082009, A082010. %Y A082006 Sequence in context: A074135
A074147 A093506 this_sequence A091451 A082330 A082329 %Y A082006
Adjacent sequences: A082003 A082004 A082005 this_sequence A082007
A082008 A082009 %K A082006 more,nonn,uned %O A082006 1,2 %A A082006
Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 05 2003 %E A082006
Needs editing - see A082002, A082003, A082004, A082005 for a model. -
njas