 # 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 ever-expanding approximation of 2^(n-p).

We saw it the previous work that, as a rule, the pth Gibonacci Sequence contains p numbers equal to 2^(n-p), 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^(n-p).

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 2n

While any Gibonacci sequence denoted by f(p, n) produces values equal to 2n-p for the range of n, p <= n <= 2p-1, the sequence begins to produce values less than 2n-p when n > 2p-1. This unto itself is not particularly interesting, until we examine the difference between a Gibonacci sequence, f(p, n) and 2n-p for values of n > 2p-1.

Now, we chart the values of the 2nd Gibonacci Sequence, f(2,n), as well as 2^(n-p) - f(2,n) : this is the difference between the "expected" value, 2^(n-p) and the actual value f(p,n), or the integer amount by which f(2,n) is less that 2^(n-p).

Consider the following chart of f(2, n), the difference between 2^(n-p) – f(2,n) and the values produced by (n+2)*2^(n-1), sequence A001792 on the AT&T/Bell Labs integer sequence site http://www.research.att.com/~njas/sequences/A001792:

 n f(2, n) 2^(n-p) – f(2,n) (n+2)*2^(n-1) 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^(n-1), 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^(n-p)

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 2n-p, 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 2n-p – (2^(n-1) x (n+2)) where n >= 3p+1:

 n f(2,n) 2^(n-p)-B (n+2)*2^(n-1) D-C n*(n+3)*2^(n-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 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 2n-p – (2^(n-1) x (n+2)) can be described by the function n*(n+3)*2^(n-3), 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^{n-p} - ((n+2) \cdot  2^{n-1}) + (n \cdot (n+3) \cdot 2^{n-3})

::/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 2n-p – (2^(n-1) x (n+2)) + (n x(n+3) x 2^(n-3)):

 N f(2,n) 2^(n-p)-B (n+2)*2^(n-1) D-C n*(n+3)*2^(n-3) F-E chebyshev 3 0 0 A001794 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:   Thermodynamics, Number Theory and The Goilden Ratio
Creation, Evolution and the Golden Rule
Theory of Order
Why Fibonacci and Gibonacci sequences appear everywhere in nature,
and how simple combinatoric math can describe how a Universe with simple beginnings evolved into a complex form