Fibonacci numbers and the Pascal Triangle

 

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TETRAHEDRAL    NUMBERS

        

        

     

These numbers correspond to placing discrete points in the cofiguration of a tetrahedron ( triangular base pyramid ). Tetrahedral numbers are pyramidal numbers and are the sum of consectuive triangular numbers. The first few are 1,4,10,20,35,56,84,120 ...The tetrahedral number is a figurate number : a number that can be represented by a regular geometric arrangement of equally spaced points.As the name suggests tetrahedral numbers can be visualised as a tetrahedron of points.

The terahedral numbers can be found in the forth diagonal of Pascal`s triangle, starting at row 4 as shown in the diagram. The first tetrahedral number is 1, the second is 4, the third is 10, the forth is 20, and so on.

     

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1

     

The sums of the consecutive triangular numbers (starting from 1) are the tetrahedral numbers.
For example : 969 = 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 + 91 + 105 + 120 + 136 + 153 or the seventeenth tetrahedral number (17 sums):

     

Numbers that are Added

The Sum

1

1

1 + 3

4

1 + 3 + 6

10

1 + 3 + 6 + 10

20

1 + 3 + 6 + 10 + 15

35

1 + 3 + 6 + 10 + 15 + 21

56

1 + 3 + 6 + 10 + 15 +21 + 28

84

1 + 3 + 6 + 10 + 15 + 21 + 28 + 36

120

1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45

165

1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55

220

     

The sum of these ... the new pattern shown in the right column ... is referenced as the sequence of tetrahedral numbers :

tetrahedral numbers

     

The tetrahedral number is a figurate number of the form:

     

     

where n= 3, 4, 5,6, ...

As it is known, the triangular number is a figurate number of the following form:

     

     

where n= 2, 3, 4, 5, ....

There is a connection with tetrahedral numbers and triangular numbers :

     

     

Now, the operator`s relation is obtained:

     

     

or:

     

     

An infinite sum of the triangular numbers as the negative tetrahedral number can be calculated, as shown in the table below :

     

Infinite series of the triangular numbers

The Sum

1+3+6+10+15+21+28+36+...

-  0

3+6+10+15+21+28+36+45+...

-  1

3+6+10+15+21+28+36+45+55+...

-  4

6+10+15+21+28+36+45+55+66+...

- 10

10+15+21+28+36+45+55+66+78+...

- 20

15+21+28+36+45+55+66+78+91+...

- 35

21+28+36+45+55+66+78+91+105+...

- 56

28+36+45+55+66+78+91+105+120+...

- 84

36+45+55+66+78+91+105+120+136+...

-120

45+55+66+78+91+105+120+136+153+...

-165

     

A tetrahedral number is the number of balls you can put in a triangular pyramid. The sum of two consecutive tetrahedral numbers is a square pyramidal number:

Tn = 35 and Tn+1 = 56


Then we have 35 + 56 = 91 which is 12 + 22 + 32 + 42 + 52 + 62,

     

as shown in the table bellow

     

tetrahedral numbers tetrahedral numbers pyramidal numbers
0
1
1
1
4
5
4
10
14
10
20
30
10
15
25
20
35
55
35
56
91
56
84
140
84
120
204

     

An interesting number`s triangle can be generated by adding up triangular numbers :

     

01 01 01 .......
3* 01
03 03 03 03 .......
3* 04
06 06 06 06 06 .......
3* 10
10 10 10 10 10 10 .......
3* 20
15 15 15 15 15 15 15 ....... 3* 35
21 21 21 21 21 21 21 21 ....... 3* 56
28 28 28 28 28 28 28 28 28 ....... 3* 84

     

Also the next number`s triangle can be generated by adding up triangular and tetrahedral numbers :

     

01 01 01 01 .......
4* 01
01 03 03 03 03 03 .......
4* 04
04 06 06 06 06 06 06 .......
4* 10
10 10 10 10 10 10 10 10 .......
4* 20
20 15 15 15 15 15 15 15 15 .......
4* 35
35 21 21 21 21 21 21 21 21 21 .......
4* 56

     

Tetrahedral numbers satisfy the recurrence relation, as shown in the table below::

     

point
line
triangle
pyramid
pyramid
1
1
1
1
4
1
2
3
4
10
1
3
6
10
20
1
4
10
20
35
1
5
15
35
56
1
6
21
56
84
1
7
28
84
120

     

4-tetrahedron numbers: 1, 5, 15, 35, 70, . . . , can be found in the fifth diagonal of Pascal`s triangle, which are of course the sum of the tetrahedral numbers.

     

     

        

        

  2001-2003 Radoslav Jovanovic                  created:  February 2003.