Note to the reader:
What follows here is a semi-rough draft of something I have been working on, off and on for the past few months. It is by no means
“camera ready.” I have written it completely in Word. I have also given on sparse comments and explanations of the mathematics
involved. I have thoroughly enjoyed looking at how the Fibonacci Sequence is not only found in Pascal’s Triangle but also looking at
it from a couple of different perspectives. You have probably seen most of what is included here, but maybe not all of it.
I would appreciate any comments or suggestions you have on the content. Email me at
[email protected]
Thanks for taking time to look at it.
The Fibonacci Sequence
And
Pascal’s Triangle
With a Little Different Slant
Pascal’s Triangle
1
1
1
1
1
1
1
1
1
8
3
5
7
6
15
Nothing new here. We’ve all seen this.
1
4
10
20
35
56
1
3
10
21
28
2
4
6
1
5
15
35
70
1
1
6
21
56
1
7
28
1
8
1
Pascal’s Triangle with Shallow Diagonals
A few shallow diagonals highlighted, note the three in colors. Note that the sums of the diagonals
create the Fibonacci Sequence.
Sum
1
1
1
1
1
1
1
1
1
8
3
5
7
6
15
4
10
20
35
56
1
3
10
21
28
2
4
6
1
5
15
35
70
1
1
2
3
5
8
1…………………………. 13
………………………. 21
1 ……………………34
1
6
21
56
1
7
28
1
8
1
Pascal’s Triangle, Reoriented
Reoriented to make the diagonals easier to see. Note the same diagonals as above are highlighted.
Sum
1
1
1
1
2
1
1
3
3
1
4
6
1
1
2
3
5
8
1 ……………………13
………………...21
4
1 ………..34
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
Shallow Diagonals and Their Sum
The diagonals are now in rows, making it a little easier to read.
1
1
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
8
1
3
6
10
15
21
1
4
10
20
1
5
1
1
2
3
5
8
13
21
34
55
Observations and questions. (Things that make you go, “Hmmmmm.”)
The number of terms in the diagonals increases by one every other row, not every row like
Pascal’s Triangle.
Is there a pattern?
Can we construct this triangle independent of Pascal’s Triangle?
What does each individual number represent?
What does it have to do with rabbits?
Pascal’s Triangle Revisited
Notation: [a,b] is the combination of a things taken b at a time. I got lazy here. It was
easier to type this notation that the normal combinatoric notation.
[0,0]
[1,0]
[2,0]
1
1
[1,1]
[2,1]
3
[2,2]
3
1
4
6
4
[a-3,1]
[a-3,2]
… …
… … …
1
[a-2,0]
[a-1,0]
[a,0]
[a-2,1]
[a-1,1]
… …
……
[a-2,2]
… …
1
Shallow Diagonals Highlighted
(Clearly Green Diagonal + Blue Diagonal = Red Diagonal.)
[0,0]
[1,0]
[2,0]
[1,1]
[2,1]
[2,2]
…
[a-3,1]
[a-2,0]
[a-1,0]
[a-2,1]
[a-3,2]
[a-2,2]
[a-1,1]
[a,0]
….
…
…
…
…
…
…
…
[n,k]
[n,k+1]
[n+1,k+1]
Shallow Diagonals and Sums
(Zeroes Added)
1
1
1
1
1
1
1
1
1
1
0
0
1
2
3
4
5
6
7
8
0
0
0
0
1
3
6
10
15
21
0
0
0
0
0
0
1
4
10
20
…….
Xa
Ya
Za
Xb
Yb
Zb
Xc
Yc
Zc
0
0
0
0
0
0
0
0
1
5
Sum
1
1
2
3
5
8
13
21
34
55
It will become clear why I added the zeroes shortly. The same three diagonals are highlighted.
The notation at the bottom gives the general postion in the grid (Xa = Row X, Column a)
We have already shown that Xa + Yb = Zb, etc. in the previous pages. We will discuss it more.
Now, I have laid out the groundwork, it is time to move to the heart of the matter.
The Question Still Remains!
What do the individual numbers in the shallow diagonals have to do with the original problem that
created the Fibonacci Sequence? Anything? Nothing? Just sheer coincidence?
Don’t forget, the Fibonacci Sequence came about as the answer to a question about rabbits and their
reproduction. Fibonacci solved the problem by counting the rabbits.
Fundamental Principle of Counting: To count correctly you must count everything once and
nothing twice.
Fibonacci method:
In order to count the number of pairs of rabbits, count the number alive last month and add to that
the number born this month. In order to give birth this month, the mother had to mate last month so
she had to be alive two months ago. So, the number of births this month is the number alive 2
months ago. In August, for example, there were 8 pairs of rabbits born. Add that to the 13 already
alive and you get 21.
This gives the recursive formula of finding any number by adding the previous two numbers.
Jan Feb Mar Apr May June July Aug …
1
1
2
3
5
8
13 21 …
But what if he had used a different strategy to count the rabbits?
Suppose he had personalized the problem so that he looked at it like this:
I will consider myself the original father. Then I will count, for every month, how many sons I
have, how many grandsons, great grandsons, and so on. I will add up those numbers for each
month and that will tell me how many pairs of rabbits there are each month.
(Note: He is going to count every pair once, and no pair twice. So, if he develops an algorithm
that is accurate, he will correctly solve the problem.)
Here is how he would have solved the problem posed to him:
Notation: I will call me Gen1, my sons Gen2, grandsons Gen3 and so on.
I know that my sons, grandsons, etc. cannot mate until they are one month old and that they will
mate every month thereafter. So, each of them will produce one son every month starting at age two
months.
Month
Gen1
Gen2
Gen3
Gen4
Gen5
Total
Jan
Feb
Mar
Apr
May
1
1
1
1
1
0
0
1
2
3
0
0
0
0
0
0
0
0
0
0
0
0
1
1
2
3
Then, he noticed a couple of definite patterns.
1. The first column will always be a 1. (Himself)
2. The second column, starting in March, will always increase by 1. (He will give birth to
another son.)
But, what about the 3rd and 4th and 5th columns? How will they get filled in?
Then he realized, “Just like me, any male who is 2 months old or older will produce a male offspring
that month and every month thereafter. So, to figure out how many grandsons I have all I have to do
is count the number of grandsons I had last month and add to it the number born this month. And,
the number born this month will be the number of sons I had two months ago. And, the same
process will work for great grandsons and so on.
So, it was easy to fill in the columns now.
Month
Gen1
Gen2
Gen3
Gen4
Gen5
Total
Jan
Feb
Mar
Apr
May
June
July
Aug
Sep
Oct
1
1
1
1
1
1
1
1
1
1
0
0
1
2
3
4
5
6
7
8
0
0
0
0
1
3
6
10
15
21
0
0
0
0
0
0
1
4
10
20
0
0
0
0
0
0
0
0
1
5
1
1
2
3
5
8
13
21
34
55
The process became simple. To find any number, just add the number directly above it to the
number above and to the left of that number. (A few are highlighted below. To find the number
boxed in red, just add the two highlighted boxes – blue, green, yellow – above it.)
Month
Gen1
Gen2
Gen3
Gen4
Gen5
Total
Jan
Feb
Mar
Apr
May
June
July
Aug
Sep
Oct
1
1
1
1
1
1
1
1
1
1
0
0
1
2
3
4
5
6
7
8
0
0
0
0
1
3
6
10
15
21
0
0
0
0
0
0
1
4
10
20
0
0
0
0
0
0
0
0
1
5
1
1
2
3
5
8
13
21
34
55
Do those numbers look familiar? They are the numbers in the shallow diagonals of Pascal’s
Triangle!
So, each number tells us how many rabbits of each generation are alive, which is more information
than we had before. All we could tell by looking at the original sequence was how many were alive
on any given month and how many of those were newborn.
But, wait! There’s more!!
Each of those numbers can be written in combinatoric form.
The numbers in the shallow diagonals are
0
( )
0
1
( )
0
2 1
( )( )
0 1
3 2
( )( )
0 1
4 3 2
( )( )( )
0 1 2
….
𝑏 𝑏−1 𝑏−2 𝑏−3
( )(
)(
)(
)…
0
3
1
2
So, we know have a non-recursive formula for the Fibonacci Sequence
Let N be the nth number in the Fibonacci Sequence. Notice that b in the above array is N - 1
Then N is equal to the following sum:
𝑁−1
∑(
𝑁−1−𝑟
)
𝑟
𝑟=0
But, wait! There is even more!!
Let {N, R} represent the number of pairs of rabbits of the Rth generation there are in the Nth month.
The Nth row in the array of the shallow diagonals is given by
𝑁−1 𝑁−2 𝑁−3
𝑁−𝑅
(
)(
)(
)…………(
)……
0
1
2
𝑅−1
Where each number tells us the number of pairs of rabbits for the Nth month and the Rth generation
are alive.
So, in the 10th month and the 4th generation, {10, 4}, we just go to the 10th row and find the 4th
number, which is
6
10 − 4
(
) = ( ) = 20
4−1
3
pairs of rabbits.
Which is exactly what the chart gives us.
Month
Oct
Gen1
1
Gen2
8
Gen3
21
Gen4
20
Gen5
5
Now, I don’t know what you think, but I think that’s pretty neat!
Total
55