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 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