[seqfan] new sequences from old
David Newman
davidsnewman at gmail.com
Thu Jan 4 20:40:22 CET 2018
I'm passing along this letter from Moshe Newman describing a method of
generating new sequences from old.
this is a general method of getting new sequences from old, provided
the old sequence is a permutation of the natural numbers.
in general, if a(n) is a sequence with no repeated values, there must
exist a shift function shift(x) with the property that
shift( a(n) ) = a(n+1). in general you can have many different such
functions, since the property that defines the shift function only
cares about its values at a(n)s.
however, if a(n) is a permutation of the natural numbers, then its
shift is unique, well defined, on the natural numbers. to be explicit,
if b(n) is the inverse permutation of
a(n), so that b(a(n)) = a(b(n)) = n, we have
shift(n) = a( b(n) + 1).
to give one explicit example, if a(n) is the greedy queens sequence:
1, 3, 5, 2, 4, 9, ...
and b(n) is the inverse
permutation : 1, 4, 2, 5, 3, 10, ...
then the shift sequence comes out to be : 3, 4, 5, 9, 2, 8, 22, ...
you could propose this for a particular sequence, and suggest to the
sequence fans that they go about systematically adding these to the
database. if you need a motive for putting these in, point out the
obvious: if by some miracle you can compute the shift of a sequence
easily, then you can compute the sequence easily by using recursion.
you can compute a(n+1) just by knowing a(n), without even knowing what
n is.
comment: the shift function is very nearly a permutation itself,
except that the value a(1) does not appear in the image.
of course, if a nonperiodic sequence has repeated values, then it
can't have a shift function. even so, it is possible that no
consecutive pair repeats, that is to say the pair of numbers ( a(n),
a(n+1) ) is not equal to the pair of numbers ( a(m), a(m+1) ),
unless of course m=n. in this case there is two-variable shift
function, such that
shift ( a(n), a(n+1) ) = a(n+2).
even if the sequence has no repeated elements, you still might want a
two-variable shift function, which may be much simpler that the
one-variable shift. see eg the fibonacci sequence.
of course, you know i have a fondness for this sort of problem, having
had one spectacular success in finding shift functions. in general, to
be sure, there is nothing nice to find.
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