Recursive-Algorith To Transform Sequences

[Christian G. Bower]

I've played around with Leroy's transform some and I have an explanation
for some of the resulting sequences:

Leroy Quet <qq-quet at mindspring.com> wrote:

> 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.)

If I take the sequence whose g.f. is:
(1+x)(1+x^2)(1+x^3)(1+x^5)(1+x^6)(1+x^8)(1+x^11)...(1+x^a(oo,n))...

I get:
1 1 1 2 1 2 3 2 4 4 3 6 4 5 8 5 8 9 7...
Taking the partial sums I get:
1 2 3 5 6 8 11 13 17 21 24 30 34 39 47 52 60 69 76...
the original sequence.

Similarly:
> 
> 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.)

(1+x)(1+x^3)(1+x^5)(1+x^8)... gives
1 1 0 1 1 1 1 0 2 2 0 1 2 2 1 1 2 3 2 0 3...
when I convolve it with 1 2 3 4 5 6...
(note I'm starting with index 0, so this is the sequence b(n)=n+1)
I get: 1 3 5 8 12 17 23 29 37 47 57 68 81 96 112...

So Leroy's transform appears to take sequence a to a sequence b such that
if I take the Weigh transform of the characteristic function of b and
convolve it with sequence a, I get sequence b.

You can view Weigh transform of characteristic function as the number
of ways to write n as the sum of distinct members of b or a partition
into distinct parts where the parts are in the range of b.

This is reminiscent of the partition transform mentioned in
http://www.research.att.com/~njas/doc/eigen.pdf
A kind of distinct parts transform.
With Leroy's transform giving a sequence stable under this transform
followed by a convolution with the input sequence.

Christian