Tribonacci related sequences.

[Creighton Dement]

Dear Seqfans, 

The sequence 1lesforcycrokseq: 1, -2, 5, -11, 26, -53, 104, -198, 375, 
caught my eye (see http://www.crowdog.de/UnknownTrib.html , noting that
les(...)seq + jes(...)seq + tes(...)seq = 0.). That is -at least partly-
because of the long name which, in a way, reflects the "relative
complexity" of the process which generated the sequence (but not
necessarily the "complexity" of the resulting sequence itself.). 

It is quite normal for me not to have any luck with SuperSeeker on
sequences of this type. Indeed, I didn't have any luck with this
sequence.  However, here are two points to consider:  a(n+1)/a(n)
appears to approach the Tribonacci Constant 1,839... 
Second, if we take away the "cyc-symmetry" (based on applying a cyclic
transformation i->j, j->k, k->i) we get  1lesforrokseq:  1, -3, 8, -18,
38, -76
( see http://www.crowdog.de/TribKnown.html ) 

with SuperSeeker results:  

%S A000001 1, 3, 8, 18, 38, 76, 147, 279, 523, 973, 1802, 3328, 6136,
11302, 20805, 38285, 70437, 129575, 238348, 438414, 806394, 1483216,
2728087, 5017763, 9229135, 16975057, 31222030
%N A000001 G.f. (1+x^2+x^4)/((x^3+x^2+x-1)(x-1)^2)
%C A000001 A floretion-generated sequence involving Tribonacci numbers.
%H A000001 C. Dement, <a
href="http://www.crowdog.de/20801/home.html">Force
Transforms</a>.
%H A000001 C. Dement, <a
href="http://www.crowdog.de/13829/home.html">The
Floretions</a>.
%F A000001 a(n+2) - 2*a(n+1) + a(n) = A081172(n+4)
%o A000001 Floretion Algebra Multiplication Program, FAMP Code:
1tesforrokseq[A*B] =
A =  - .5'ii' + .5'jj' + .5'kk' + .5e
B =  + 'kj', 1vesforrokseq[A*B] = A000004, ForType: 1A.
%Y A000001 Cf. A081172.
%O A000001 0
%K A000001 ,nonn,
%A A000001 Creighton Dement (crowdog at crowdog.de), Apr 01 2005


Is it possible for someone to find the g.f. for the first case, above?
Note: Though it looks rather unlikely by the sequence's orderly
appearance, (refer to my previous seqfan message "Initial terms..."), it
is still quite possible that initial term(s) may  have to be left off in
order to find the "appropriate" generating function. 

Sincerely, 
Creighton