Two pleasant and easygoing sequences with monster generating functions.

[creigh at o2online.de]

Dear Seqfans,

A newly defined (dynamic symmetry) "evenseq" sums over all floretion basis vectors 
with coefficients whose integer part (floor) is an even number.   (I will 
try to add this to an update of FAMP later this evening)

The floretion - 0.25'i + 0.5'k - 0.25i' - 0.5j' + 0.5k' - 0.75'ii' + 0.75'jj' -
 0.25'kk' + 0.25'jk' - 0.5'ki' + 0.25'kj' + 0.25e 
then generates the sequence

evenseq: 0, 1, 1, 0, 0, 5, 8, 13, 21, 0, 0, 89, 144, 233, 377, 0, 0, 1597, 2584, 
4181, 6765, 0,  

What does Superseeker have to say about that?

-x^9-x^8-2*x^7-7*x^5+x^4-x^3-x+(-2*x^8-x^10-x^9-3*x^7-5*x^6-8*x^5+5*x^4-3*x^3+2 
*x^2-x+1)*F(x) = 0 

If this is correct the next 6 numbers in the sequence are: 

[0, 28657, 46368, 75025, 121393, 0] 

The next terms are correct, but what an ugly generating function! (am I 
missing something here? Of course, beauty is in the eye of the beholder.); 
superseeker had nothing left to say about the sequence.

Another example (submitted yesterday)

I A100212 
%S A100212 0, 0, 0, 0, 8, 20, 24, 8, 0, 0, 0, 0, 128, 320, 384, 128, 0, 
0, 0, 0, 2048, 5120, 6144, 2048, 0, 0, 0, 0, 32768, 81920, 98304 
%C A100212 [[[ Note: this sequence is apparently related to A038503, Sum 
of every 4th entry of row n in and Pascal's triangle and A009116, Expansion 
of cos(x)/exp(x) via a set of new sequences which has not yet been submitted. 
]]] 

%H A100212 C. Dement, <a href="http://www.crowdog.de/13829/home.html">The 
Floretions</a>. 
%F A100212 G.f. (x^5+2*x^4)/(.5*x^2-2*x^6+2*x^5-x^4-.5*x+.25) 
(a(n)) = negseq(.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e) 

a(8n+4) = a(8n+7) = 2^(4n+3) 
a(8n+5) = (5/2)*2^(4n+3) 
a(8n+6) = 3*2^(4n+3) 
a(8n+8) = 0 
a(8n+9) = 0 
a(8n+10) = 0 
a(8n+11) = 0 
%o A100212 Floretion Algebra Multiplication Program, FAMP 
%Y A100212 C.f. A100213, A038503, A009116 
%O A100212 0 
%K A100212 ,nonn, 
%A A100212 Creighton Dement (creighde at o2online.de), Nov 08 2004 
RH 

Actually, the g.f., from above, was origianlly given by Guesss as 
-1/412*x^5-1/206*x^4+(1/824*x^2-1/206*x^6+1/206*x^5-1/412*x^4-1/824*x+1/1648)*F 
(x) = 0 

I apologize for not submitting all sequences mentioned in the note (enclosed in 
brackets) at once, but I work as an English teacher during the week :) . 

Sincerely, 
Creighton