[seqfan] Timary Sequence

[Quaid Carlo Bulloch]

Hello to whom it may concern,

I've found a sequence I'd like to share with you!

Similar to this period-doubling tree :: H,L:HH,HL,LH,LL

The set I found always maintains the initial element size as the set grows:
T2[H,L] =   H,L: HL,LH: HLLH,LHHL:
HLLHLHHL,LHHLHLLH: HLLHLHHLLHHLHLLH, LHHLHLLHLHHLHLLH: ....
In the above case, there will always be two elements per layer. There
are many cool things about this sequence that include nesting and
conversion to integer sequences (In particular, binary to base ten
sequences).

"T" of some set takes the first element of said set and concatenates it
with the second, and vice versa. The subscript two represents the number of
elements read in each instance, which may be equal to but not exceed the
set size.

(e.g)
T1[H,L,K] = H,L,K: H,L,K: H,L,K:...
T2[H,L,K] = H,L,K: HL,LK,KH: HLLK,LKKH,KHHL:...
T3[H,L,K] = H,L,K: HLK,LKH,KHL: HLKLKHKHL, LKHKHLHLK, KHLHLKLKH:...

Nesting requires two rules but it has lots of potentials:
For T2(x), starting on the rightmost index, take the product of any term on
the right of a T(set). Replace "x" with the left term's index
representation in the x set, where "x"=set contents.

(e.g) 1.1:
 T2(1,0,*T2(1,0,1)*) (Base 2): 10,0T2(x),T2(x)1: 100T2(x),0T2(x)T2(x)1, T2
(x)110:...

Internal T set expansion, first term is index =0, last term is index = 5 :
*T2(1,0,1)* = 10,01,11: 1001,0111,1110

Going back to the original set [1.1], the second term is 0*T2(x), since "0"
is on the left of the T set we grab the zeroth index of this internal T =>
"10" yielding: "10,10" so far.
The next element in this layer is 1*T2(x)1 which is the first element in
the internal T set => one times  "01" = "01" yielding "10,10,01" so far.

Further expansion according to these rules: 10,10,01:011,11,110:11, 110110
and so on.

If we put any T2 set into a binary sequence then convert it to base ten we
get the following sequences:

T2[0,1,1] (Base 2) = 01,11,10: 0111,1110,1001: 01111110,11101001,10010111 =
(Base --> 10) = 1,3,2,7,14,9,126, 233, 151, 32489...
T2[1,0,1] (Base 2 --> 10) =
1,0,1,3,1,2,14,7,9,233,126,151,59799,32489,38782...
T2[1,0] (Base 2 --> 10) = 0,1,1,2,6,9,150,105,38505,27030...

Universally in a T set there exist a tepid growth depending on the size of
the initial set followed by rapid expansion, potential uses include
efficient sigmoidal functions and set expansion towards analogous
interpretations.

Alternatively, representations are attached.


Best,


Quaid Carlo Bulloch