# Recursive-Algorith To Transform Sequences

Leroy Quet qq-quet at mindspring.com
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,...
=a(1,k):1,2,2,2,2,2,2,2,2,2,...
+       0,0,1,2,2,2,2,2,2,2,...
=a(2,k):1,2,3,4,4,4,4,4,4,4,...
+       0,0,0,1,2,3,4,4,4,4,...
=a(3,k):1,2,3,5,6,7,8,8,8,8,...
+       0,0,0,0,0,1,2,3,5,6,...
=a(4,k):1,2,3,5,6,8,10,11,13,14,...
+       0,0,0,0,0,0,1, 2, 3, 5,...
=a(5,k):1,2,3,5,6,8,11,13,16,19,...
+       0,0,0,0,0,0,0, 0, 1, 2,...
=a(6,k):1,2,3,5,6,8,11,13,17,21,...

(The _numbers_ of leading 0's in the sequence following each + forms the
limit-sequence.)

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:
1,2,3,5,6,8,11,13,17,21,...

(Not in EIS.)

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:

2,3,3,2,5,4,4,0,3,...?

We get:
2,3,5,7,11,13,17,19,23,...
the beginning of the prime sequence.

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

transforms at all.
(You want to make sure that eventually a limit sequence is reached.)