Recursive-Algorith To Transform Sequences

Leroy Quet qq-quet at
Fri Jun 11 22:43:06 CEST 2004

[crossposted to sci.math]

Here is an idea for a transform of positive integer sequences.

I wonder what the closed forms would be for such transformed sequences 
derived originally from simple sequences (such as 1,1,1,1,1... and 
1,2,3,4,5,...) or the closed form of the inverse transform applied to 
simple sequences (such as the primes).

Let a(0,k) be the starting integer sequence.
(First term is a(0,1).)

Let, for m =  positive integer,

a(m,k) = a(k,m-1) + a(m-1,k-a(m-1,m)) for k > a(m-1,m);
a(m,k) = a(k,m-1) for k <= a(m-1,m).

So, for example:

a(0,k): 1,1,1,1,1,1,1,1,1,1,...
+       0,1,1,1,1,1,1,1,1,1,...
+       0,0,1,2,2,2,2,2,2,2,...
+       0,0,0,1,2,3,4,4,4,4,...
+       0,0,0,0,0,1,2,3,5,6,...
+       0,0,0,0,0,0,1, 2, 3, 5,...
+       0,0,0,0,0,0,0, 0, 1, 2,...

(The _numbers_ of leading 0's in the sequence following each + forms the 

The limit-sequence shares its first 10 terms with a(6,k), so the limit 
sequence (the sequence {a(m,k)} as m -> oo) begins:

(Not in EIS.)

If we instead start with
a(0,k) =k,
we get the limit sequence beginning:
1, 3, 5, 8, 12, 17, 23, 29, 37,...

(Not in EIS either.)

(We can, of course, even let a particular limit-sequence => another 
{a(0,k)}, transforming a sequence more than once.)

What happens if we transform the sequence:


We get:
the beginning of the prime sequence.

(And the inverted transform on the prime sequence is not in the EIS 

I have not thought about this much, such as which source sequences have 
transforms at all.
(You want to make sure that eventually a limit sequence is reached.)

Anything anyone wants to add?

Leroy Quet

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