Orloj clock sequences

franktaw at netscape.net franktaw at netscape.net
Tue Apr 25 23:09:07 CEST 2006

I got interested in the Orloj clock sequence, A028355; specifically, in 
how to generalize it.

The repeating sequence used here, 1,2,3,4,3,2, has two properties: one 
can generate the natural numbers from summing consecutive values in the 
sequence, and it goes up from 1 to 4 and then back down again.  We 
can't duplicate both properties except for the trivial case 1,2.  But 
we can duplicate the other property.

If we take the cumulative sums of the sequence 
(1,3,6,10,13,15,16,18,21,25,28,...), it must contain every triangular 
number; this is obviously a necessary and sufficient condition.  
Another way to state this is that the cumulative sums of the repeating 
sequence must contain every triangular number modulo the sum of the 
repeating sequence.

So start with a number n, determine all triangular residues mod n, sort 
them, take their differences, and voila.  It turns out that if n is 
even, the result is just the sequence for n/2 repeated twice, so take n 
odd.  The Orloj sequence is this procedure applied to n = 15.

For example, n=21, the residues are 0,1,3,6,10,15,0,7,15,3,13,3,?; 
sorted and unique we get 0,1,3,6,7,10,13,15[,21], and the differences 
give us the sequence 1,2,3,1,3,3,2,6.

Given a primitive sequence of this type, additional sequences can be 
generated by repeating it some number of times, and by refining it by 
splitting some of the numbers.  For example, we could refine the above 
sequence to 1,2,3,1,3,3,2,1,5.  Every sequence with this property can 
be generated by repeating and then refining some primitive sequence.

Franklin T. Adams-Watters

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