 # 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.replica rolex replica replica watches

 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 2n-p. In fact, as a general rule, for any Gibonacci sequence, the sequence given by f(p, n) will contain p elements equal to 2n-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 2n-p.

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