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TETRAHEDRAL NUMBERS
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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 :
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.
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