[seqfan] A new sequence

Richard Guy rkg at cpsc.ucalgary.ca
Tue Feb 5 16:41:16 CET 2013


Dear all,
          Using an old address, hoping to attract attention.
Although in theory an editor, I don't have the competence to
submit sequences properly, and I haven't persuaded anyone to
put in the thousand or so divisibility sequences that ought
to be in OEIS.  For example, there are no fewer than five
recurring sequences with characteristic polynomial

x^4 - 40x^3 + 206x^2 - 40x + 1, which have initial values

(1), 0, 1, 48, ...  (-1), 0, 1, 6, ...  (-1), 0, 1, 30, ...
                     (-1), 0, 1, 34, ... (-1), 0, 1, 48, ...

and are divisibility sequences.

I haven't checked whether any or all of these are in OEIS,
but the following sequence, which is NOT a div seq, is not.

It satisfies the same recurrence relation, and has a good
deal more mathematics in it than most of the recent submissions
that I've seen.

(-1), 0, 1, 2, 595, 19720, 667029, 22642620, 769085031,
26125682960, 887500839785, ...

It displays some of the ``ranks of apparition'' properties
that div seqs possess, and it will be seen that any three
consecutive terms have quite a large gcd:

1, 1, 5, 17, 29, 33, 169, 577, 985, ...

[The OEIS loses a good deal by not giving factorizations of
those sequences where this is a significant property.  I'll
give the relevant ones here in the hope of whetting curiosity:

595 = 5*7*17, 19720 = 2^3*5*17*29, 667029 = 3*11*17*29*41,
22642620 = 2^2*3*5*7*11*13^2*29, 769085031 = 3*11*13^2*239*577,
26125682960 = 2^4*5*13^2*17*197*577,
887500839785 = 5*7*19*59*197*199*577, ...]

Perhaps someone can explain that in terms of Lucas functions?

Hint: the roots of the characteristic polynomial are  a^k
with  k = 4, -4, 2, -2,  and  a = 1 + sqrt(2).

Anyone awake out there?   Best wishes!    R.



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