New Territory
To this point, I have discussed how the Gibonacci sequences extend the logic of the Fibonacci Sequence, and in doing so, produce an everexpanding approximation of 2^(np).
We saw it the previous work that, as a rule, the pth Gibonacci Sequence contains p numbers equal to 2^(np), according to this site's conventions.
So I asked another, rather simple question: What about all of the other numbers in Gibonacci Sequences not equal to 2^(np).
I am not really sure of the semantics behind this line of inquiry, but the love of playing with numbers seems value enough.
Gibonacci Sequences after 2^{n}
While any Gibonacci sequence denoted by f(p, n) produces values equal to 2^{np} for the range of n, p <= n <= 2p1, the sequence begins to produce values less than 2^{np} when n > 2p1. This unto itself is not particularly interesting, until we examine the difference between a Gibonacci sequence, f(p, n) and 2^{np} for values of n > 2p1.
Now, we chart the values of the 2nd Gibonacci Sequence, f(2,n), as well as 2^(np)  f(2,n) : this is the difference between the "expected" value, 2^(np) and the actual value f(p,n), or the integer amount by which f(2,n) is less that 2^(np).
Consider the following chart of f(2, n), the difference between 2^(np) – f(2,n) and the values produced by (n+2)*2^(n1), sequence A001792 on the AT&T/Bell Labs integer sequence site http://www.research.att.com/~njas/sequences/A001792:
n 
f(2, n) 
2^(np) – f(2,n) 
(n+2)*2^(n1) 
0 
0 


1 
1 


2 
1 


3 
2 


4 
3 
1 
1 
5 
5 
3 
3 
6 
8 
8 
8 
7 
13 
19 
20 
8 
21 
43 
48 
9 
34 
94 
112 
10 
55 
20i1 
256 
This third function, (n+2)*2^(n1), does not have a natural purpose here. This sequence was found by searching the AT&T Bell labs Integer Sequence site. For p=2, this new function provides three terms which, when added to f(p,n) equal to 2^(np)
Here we can see that for values of n between 2p <= n <= 3p:
1.10
Nonetheless, once n > 3p, this additional function fails to produce the proper values for the difference between f(p, n) and 2^{np}, subtracting incrementally greater values than is required to maintain the equality. So we repeat our inquisition into this system and consider the differences between f(p, n) and 2^{np} – (2^(n1) x (n+2)) where n >= 3p+1:
n 
f(2,n) 
2^(np)B 
(n+2)*2^(n1) 
DC 
n*(n+3)*2^(n3) 
0 
0 




1 
1 




2 
1 




3 
2 




4 
3 
1 
1 


5 
5 
3 
3 


6 
8 
8 
8 


7 
13 
19 
20 
1 
1 
8 
21 
43 
48 
5 
5 
9 
34 
94 
112 
18 
18 
10 
55 
201 
256 
55 
56 
11 
89 
423 
576 
153 
160 
12 
144 
880 
1280 
400 
432 
Once again, we see that p+1 (3) values in the sequences of differences between f(p, n) and 2^{np} – (2^(n1) x (n+2)) can be described by the function n*(n+3)*2^(n3), A001793, http://www.research.att.com/~njas/sequences/A001793. This results in the following equality:
1.11
::eq::
3p+1 \leq n \leq 4p+1 : f(p, n) = 2^{np}  ((n+2) \cdot 2^{n1}) + (n \cdot (n+3) \cdot 2^{n3})
::/eq::
We will perform this operation one last time before considering values of p>2, by considering the sequence of numbers produced by the difference between f(2, n) and 2^{np} – (2^(n1) x (n+2)) + (n x(n+3) x 2^(n3)):
N 
f(2,n) 
2^(np)B 
(n+2)*2^(n1) 
DC 
n*(n+3)*2^(n3) 
FE 
chebyshev 3 
0 
0 






1 
1 






2 
1 






3 
2 






4 
3 
1 
1 




5 
5 
3 
3 




6 
8 
8 
8 




7 
13 
19 
20 
1 
1 


8 
21 
43 
48 
5 
5 


9 
34 
94 
112 
18 
18 


10 
55 
201 
256 
55 
56 
1 
1 
11 
89 
423 
576 
153 
160 
7 
7 
12 
144 
880 
1280 
400 
432 
32 
32 
13 
233 
1815 
2816 
1001 
1120 
119 
120 
14 
377 
3719 
6144 
2425 
2816 
391 
400 
15 
610 
7582 
13312 
5730 
6912 
1182 
1232 
Performing the difference operation produces another sequence of numbers that does have an equational representation, however it is lengthy and, more to the point, not found in the AT&T/Bell Labs online integer sequence database as an expression, but rather are a sequence of numbers that provide roots to a Chebyshev Polynomial.
To further expound on this point, we have included below the same chart for the first three Gibonacci sequences, f(2,n), f(3,n) and f(4,n). Examining the three consecutively, we begin to see a greater pattern of logic emerging, most notably related to the pth Gibonacci sequence requiring p+1 terms of each Chebyshev function. see below:replica panerai watch bands rolex watch imitation