Gibonacci Sequences
Gibonacci is a series of sequences of numbers.
Mathematically, f(p,n) represents the nth number of the pth Gibonacci sequence. We have previously defined that f(p,n) is the sum of the previous p terms. Mathematically, this is written as:
1.3
More formally, in summation notation:
1.4
For the Fibonacci Sequence we add the two previous values in the sequence to produce the next, therefore in Gibonacci notation, p=2 and the Fibonacci sequence, f(2,n) is expressed as
1.5
For posterity, this produces the following sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ...
The next sequence in the Gibonacci series is p=3, f(3,n) is defined as
1.6
and this produces:
0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81...
This is followed by the Gibonacci Sequence p=4, f(4, n):
1.7
which generates:
0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108...
and so on, as p expands towards infinity.
The numbers produced by the Gibonacci series
While speaking of how the Gibonacci Series of Sequences works, it is a far more powerful didactic to see the values the function produces. Below is a graph of the first 12 Gibonacci Sequences; we will very quickly see a startling pattern in the values this series produces.
n | f(1,n) | f(2,n) | f(3,n) | f(4,n) | f(5,n) | f(6,n) | f(7,n) | f(8,n) | f(9,n) | f(10,n) |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
2 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
3 |
1 |
2 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
4 |
1 |
3 |
2 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
5 |
1 |
5 |
4 |
2 |
1 |
1 |
0 |
0 |
0 |
0 |
6 |
1 |
8 |
7 |
4 |
2 |
1 |
1 |
0 |
0 |
0 |
7 |
1 |
13 |
13 |
8 |
4 |
2 |
1 |
1 |
0 |
0 |
8 |
1 |
21 |
24 |
15 |
8 |
4 |
2 |
1 |
1 |
0 |
9 |
1 |
34 |
44 |
29 |
16 |
8 |
4 |
2 |
1 |
1 |
10 |
1 |
55 |
81 |
56 |
31 |
16 |
8 |
4 |
2 |
1 |
11 |
1 |
89 |
149 |
108 |
61 |
32 |
16 |
8 |
4 |
2 |
12 |
1 |
144 |
274 |
208 |
120 |
63 |
32 |
16 |
8 |
4 |
13 |
1 |
233 |
504 |
401 |
236 |
125 |
64 |
32 |
16 |
8 |
14 |
1 |
377 |
927 |
773 |
464 |
248 |
127 |
64 |
32 |
16 |
15 |
1 |
610 |
1705 |
1490 |
912 |
492 |
253 |
128 |
64 |
32 |
16 |
1 |
987 |
3136 |
2872 |
1793 |
976 |
504 |
255 |
128 |
64 |
17 |
1 |
1597 |
5768 |
5536 |
3525 |
1936 |
1004 |
509 |
256 |
128 |
18 |
1 |
2584 |
10609 |
10671 |
6930 |
3840 |
2000 |
1016 |
511 |
256 |
19 |
1 |
4181 |
19513 |
20569 |
13624 |
7617 |
3984 |
2028 |
1021 |
512 |
20 |
1 |
6765 |
35890 |
39648 |
26784 |
15109 |
7936 |
4048 |
2040 |
1023 |
21 |
1 |
10946 |
66012 |
76424 |
52656 |
29970 |
15808 |
8080 |
4076 |
2045 |
22 |
1 |
17711 |
121415 |
147312 |
103519 |
59448 |
31489 |
16128 |
8144 |
4088 |
Highlighted in bold, we see that each and every Gibonacci Sequence has a certain number of values equal to 2^{n-p}. In fact, as a general rule, for any Gibonacci sequence, the sequence given by f(p, n) will contain p elements equal to 2^{n-p} and those elements are located p <= n <= 2p-1. To denote this formally, we say that:
1.8
We can assert something very interesting about the nature and purpose of the Gibonacci sequences; Gibonacci sequences are an ever-expanding, recursive approximation of 2^{n-p}.