[seqfan] More sequences
Richard Guy
rkg at cpsc.ucalgary.ca
Sun May 9 19:05:58 CEST 2010
Here's an(other) idea for infinitely many sequences.
It arose from a quest for chains and necklaces of
positive integers, the sum of any consecutive pair
of which is a perfect square.
1,3,6,10,15,21,4,5,11,14,2,7,9,16,20,29,35,46,18,31,33,48,52,...
(I've probably made enough errors by now)
The idea was: a(1) = 1, (can include a(0) = 0, but in
some other cases this causes trouble) and a(n) for
n > 1 is the least positive integer not already in the
sequence such that a(n) + a(n-1) is a perfect square.
Some questions arise:
Are there any values of n other than 1 such that
the first n members of the sequence comprise the
numbers 1 to n ?
Are there any positive integers which do not occur in
the sequence?
Instead of the squares, one can use any other sequence
for the base sequence, provided it contains arbitrarily
large members.
If the base sequence is the even numbers, the result is
the odd numbers. If the base sequence is the odd numbers
the result is the natural numbers. It's doubtful if this
merits a comment at either of the last two.
If the base sequence is the Fibonacci numbers, then the
result is the Fibonacci numbers. If that's correct it
might merit a comment.
If the base sequence is the Lucas numbers, 2,1,3,4,7,11,...
then we get (I believe)
1,2,5,6,12,17,30,46,77,122,200,321,522,...
whose members are alternately one less and one more than
the Lucas numbers themselves, L_n - (-1)^n (n > 1).
Perhaps that's already been noticed ?
If the base sequence is the triangular numbers, then we
get (E&OE as usual)
1,2,4,6,9,12,3,7,8,13,15,21,24,31,35,10,5,16,20,25,11,17,...
Is there an enthusiastic editor who would like to submit
a few, a lot, of these? R.
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