Download How Pascal`s Triangle is Constructed

How Pascal's Triangle is Constructed
At the tip of Pascal's Triangle is
the number 1, which makes up
the zeroth row. The first row (1
& 1) contains two 1's, both
formed by adding the two
numbers above them to the left
and the right, in this case 1 and 0
(all numbers outside the Triangle
are 0's). Do the same to create the
2nd row: 0+1=1; 1+1=2; 1+0=1.
And the third: 0+1=1; 1+2=3;
2+1=3; 1+0=1. In this way, the
rows of the triangle go on
infinitly. A number in the triangle
can also be found by nCr (n
Choose r) where n is the number
of the row and r is the element in
that row. For example, in row 3,
1 is the zeroth element, 3 is
element number 1, the next three
is the 2nd element, and the last 1
is the 3rd element. The formula
for nCr is:
n!
-------r!(n-r)!
! means factorial, or the preceeding number multiplied by all the positive integers that are
smaller than the number. 5! = 5 × 4 × 3 × 2 × 1 = 120.
The Sums of the Rows
The sum of the numbers in any row is equal to 2 to the nth power or 2n, when n is the
number of the row. For example:
20 = 1
21 = 1+1 = 2
22 = 1+2+1 = 4
23 = 1+3+3+1 = 8
24 = 1+4+6+4+1 = 16
Prime Numbers
st
If the 1 element in a row is a prime number (remember, the 0th element of every row is
1), all the numbers in that row (excluding the 1's) are divisible by it. For example, in row
7 (1 7 21 35 35 21 7 1) 7, 21, and 35 are all divisible by 7.
Hockey Stick Pattern
If a diagonal of numbers of any
length is selected starting at any
of the 1's bordering the sides of
the triangle and ending on any
number inside the triangle on that
diagonal, the sum of the numbers
inside the selection is equal to the
number below the end of the
selection that is not on the same
diagonal itself. If you don't
understand that, look at the
drawing.
1+6+21+56 = 84
1+7+28+84+210+462+924 =
1716
1+12 = 13
Magic 11's
If a row is made into a single number by using each element as a digit of the number
(carrying over when an element itself has more than one digit), the number is equal to 11
to the nth power or 11n when n is the number of the row the multi-digit number was taken
from.
Row # Formula = Multi-Digit number
Actual Row
0
Row 0 11
=
1
1
1
Row 1 11
=
11
11
2
Row 2 11
=
121
121
113
114
115
116
117
118
Row 3
Row 4
Row 5
Row 6
Row 7
Row 8
=
=
=
=
=
=
1331
14641
161051
1771561
19487171
214358881
1331
14641
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
Fibonnacci's Sequence
Fibonnacci's Sequence can also be located in
Pascal's Triangle. The sum of the numbers in the
consecutive rows shown in the diagram are the first
numbers of the Fibonnacci Sequence. The Sequence
can also be formed in a more direct way, very
similar to the method used to form the Triangle, by
adding two consecutive numbers in the sequence to
produce the next number. The creates the sequence:
1,1,2,3,5,8,13,21,34, 55,89,144,233, etc . . . . The
Fibonnacci Sequence can be found in the Golden
Rectangle, the lengths of the segments of a
pentagram, and in nature, and it decribes a curve
which can be found in string instruments, such as
the curve of a grand piano. The formula for the nth
number in the Fibonnacci Sequence is
Try the formula out below (requires JavaScript)
Enter N:
Result:
Triangular Numbers
Triangular Numbers are just one
type of polygonal numbers. See
the section on Polygonal
Numbers for an explaination of
polygonal and triangular
numbers. The triangular numbers
can be found in the diagonal
starting at row 3 as shown in the
diagram. The first triangular
number is 1, the second is 3, the
third is 6, the fourth is 10, and so
on.
Square Numbers
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 4-sided 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.
Polygonal Numbers
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
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
Find the nth x-gonal number using the Winton formula:
N:
X:
Result:
Find the nth x-gonal number using the Shi-Cheng formula:
N:
X:
Result:
Type
1
st
2nd
3rd
4th
5th
6th
7th
Triang
ular
Value 1
Square
3
6
10
15
21
28
Value 1
Pentag
onal
4
9
16
25
36
49
Value 1
Hexag
onal
5
12
22
35
51
70
Value 1
6
15
28
45
66
91
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 rightangled form of Pascal's Triangle, except the preceeding 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.
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.