Download Square Numbers

All You Ever Wanted to Know about Pascal’s Triangle – and more!
Square Numbers
http://ptri1.tripod.com/#polygonal
Square Numbers are another type of Polygonal
Numbers.
They are found in the same diagonal as the
triangular numbers.
A Square Number is the sum of the two numbers in
any circled area in the diagram. (The colors are
different only to distinguish between the separate
"rubber bands"). The nth square number is equal to
the nth triangular number plus the (n-1)th triangular
number. (Remember, any number outside the
triangle is 0). The interesting thing about these 4sided polygonal numbers is that their name explains
them perfectly. The very first square number is 02.
The second is 12, the third is 22 (4), the fourth is 32
(9), and so on. Read on to the Polygonal Number
section to learn more.
Connection to Sierpinski's Triangle
When all the odd numbers
(numbers not divisible by 2) in
Pascal's Triangle are filled in
(black) and the rest (the evens)
are left blank (white), the
recursive Sierpinski Triangle
fractal is revealed (see figure at
near right), showing yet another
pattern in Pascal's Triangle.
Other interesting patterns are
formed if the elements not
divisible by other numbers are
filled, especially those indivisible
by prime numbers. Go here to
download programs that calculate
Pascal's Triangle and then use it
to create patterns, such as the
detailed, right-angle Sierpinski
Triangle at the far right.
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Points on a Circle
Image
Points Segments Triangles Quadrilaterals Pentagons Hexagons Heptagons
1
2
1
3
3
1
4
6
4
1
5
10
10
5
1
6
15
20
15
6
1
7
21
35
35
21
7
1
As you may have noticed, the numbers in the chart above are actually the tip of the right-angled form of
Pascal's Triangle, except the preceding 1's in each row are missing. The circular figures are formed by simply
placing a number of points on a circle and then drawing all the possible lines between them. This chart shows
that for a figure with n points, all you need to do is look at the nth row of the Triangle in order to find the
number of points, line segments, and polygons in the figure with ALL of their vertices on the circle.
Polygonal Numbers
http://ptri1.tripod.com/#polygonal
Polygonal Numbers are really just the number of vertexes in a figure formed by a certain polygon. The first
number in any group of polygonal numbers is always 1, or a point. The second number is equal to the number
of vertexes of the polygon. For example, the second pentagonal number is 5, since pentagons have 5 vertexes
(and sides). The third polygonal number is made by extending two of the sides of the polygon from the second
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polygonal number, completing the larger polygon, and placing vertexes and other points where necessary. The
third polygonal number is found by adding all the vertexes and points in the resulting figure. (Look at the table
below for a clearer explaination). My colleagues' formula (the Shi-Cheng formula) for the nth x-gonal number
(for example: the 2nd 3-gonal, or triangular number) is:
If x is even, then:
y = x/2 - 1 and the formula is n+y(n2-n)<BRGT;
If x is odd, then:
y = (x-1)/2 and the formula is (-(n2)+3n+2n2y-2ny)/2
These formulas work fine, but I think my own formula (the Winton formula) is much less convoluted, and is
based on the fact that to find the nth x-gonal number, you multiply the number in the 3rd diagonal in the nth row
by x-2, and then add the number in that same row but in the 2nd diagonal. Therefore:
((n2-n)/2) × (x-2) + n
Type
1st
Triangular
2nd
3rd
4th
5th
6th
7th
1
3
6
10
15
21
28
Value
1
Pentagonal
4
9
16
25
36
49
Value
1
Hexagonal
5
12
22
35
51
70
6
15
28
45
66
91
Value
Square
Value
1
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