An Exploration Of Difference Triangles

By Adrian Ferenc

April 3, 2008

Occidental College

Outline

I. Introduction II. Difference Triangles III. Increasing Triangles IV. Row Sum Triangles V. Open Questions

Axiomatically Defined Structures

Algebraic Group

Equivalence Relation

Topology

Pascal’s Triangle

11

11

11

11

11

1

cba

c

ba

Pascal’s Triangle Continued

15101051

14641

1331

121

11

1

Difference Triangles

A triangle is a Difference Triangle if it adheres to the following condition:

For any .

All entries are nonnegative integer values.

cbac

ba,

An Example

Sloane’s A059346

36 51 72 102 145 207 297 429

15 21 30 43 62 90 132

6 9 13 19 28 42

3 4 6 9 14

1 2 3 5

1 1 2

0 1

1

A Blank Triangle

1,1

2,22,1

3,33,23,1

4,44,34,24,1

c

cc

ccc

cccc

jijiji ccc ,1,11,

Removing any number of the end columns still results in a difference triangle.

Multiplying our triangle by a “scalar” results in a difference triangle.

If in a triangle there exists at least one element in every row that is divisible by some k, then every element of the triangle is divisible by k.

Some Facts About Difference Triangles

Increasing TrianglesIf in a difference Triangle, for all

Or, equivalently, in any row m,

Then this triangle is an Increasing triangle.

cabc

ba,

mmmm ccc ,,2,1

Increasing Triangles Continued

In the same way Pascal’s Triangle was constructed by only using we can similarly construct an increasing triangle

...1,1,1,1,1

1

1

1

1

1

21

421

8421

Facts About Increasing Triangles We can add together any two increasing triangles. Since in any row m, , if we sum up

all the elements in a row we get:

which equals .

mmmm ccc ,,2,1

m

kmmmmmmmmmk ccccccc

11,1,11,21,31,11,2,

1,11,1 mmm cc

An Example

What if we choose the leftmost column to be the numbers 1, 1, 2, 3, 5, 8, …, i.e., the Fibonacci Sequence.

Extrapolating on this we can see that:

1

21

532

13853

34211385

112

12

nn

n

nkn fff

Row Sum Triangles

A Triangle is a Row Sum triangle if it is a difference

triangle and, for some infinite sequence ,

row n in the Triangle sums to, so for any row m,

,,, 321 aaa

na

m

immi ac

1,

So Why the Restriction?

Let’s remove the absolute value restriction and see what happens…

For any .cabc

ba,

cabc

ba,

Let’s have our rows sum up to the numbers 1, 1, 2, 3, 5, etc…, i.e., the Fibonacci numbers:

Row 1Rows 1 & 2

Rows 1 – 3

So there is no integral solution.

1

1

10

1

10

yxx

34

43

1

22

y

y

xy

yx

Example 1

Let’s have our row sum sequence be 1,1,1,…

Increasing Unique Multiply by k 1

10

100

1000

“Example 2”

What if we try the sequence 1,2,3,4,…

For row 1 we get 1.

For row 2 we get x, y where x + y = 2, x - y = 1.

Clearly Nyx,

Nonexistent Row Sum Triangles

So for such a sequence to correspond to a row sum triangle.

Also Similarly, our sequence cannot be infinitely decreasing.

)2(mod21 aa

21 aa

)2(modyxyx

yxyx

Our row sum sequence can however “jiggle,” or increase and decrease.

1111

01010

0110

010

Example 2

Let’s try the row sum sequence 1, 3, 5, 7,…

Not an increasing triangle. Unique?

1

12

320

5200

Non-Uniqueness

1

12

320

3022

30024

300026

Example 2′ Let’s try the row sum sequence 0, 2, 4, 6,…

0

11

121

3201

52001

1

12

320

5200

Back To Pascal’s Triangle

15101051

14641

1331

121

11

1

Note that here the rows sum up to 1, 2, 4, 8, … or12 k

Example 3

Let’s have our sequence be 2, 4, 8, 16, …

2

31

521

9421

n

k

nk

1

1 122

Example 4

Let’s have our sequence be 1, 3, 9, 27, …

1

21

423

8469

n

k

nkkn

1

1)1( 323

Example 5, A Generalization

So we’ve seen and .

Are we able to generalize this to ?

n2 13 n

nN

Yes!

In fact we can do more!

We can create a row sum triangle whose rows sum to:

nn NN N

nn NN

1

1 1

1, )1(1

1

ij

k i

kjkji iN

k

ijc

We can create a row sum triangle for the above sequence by:

N

“Proof”

Outline of proof: Part 1: Show

Part 2: Show that this Triangle is increasing by showing that

This tells us that

0, yxc

1,1,1, yxyxyx ccc

1,11,11

,

nnn

n

knk ccc

Part 3: Here we show that:

Combining parts 1 – 3 we get:

“Proof” Continued

11,1

i

mm iNc

11,1 )1(

i

mmm iNc

n

k i i

nnnnnk NNiNiNc

1 1 1, )()()1(

Taste Of Proof

Example 6 Let’s try a different type of sequence. The Fibonacci Sequence is defined recursively as follows:

.1,01 2101221 ffffffff kkk

)3(mod1

)3(mod0

)3(mod2

3

2

1

,

ijiff

ijiff

ijiff

c

i

i

i

ji

The Fibonacci Triangle

Fibonacci Triangle Continued Is it unique? No

Since 12 kkk fff

Back To Pascal’s Triangle

Zalman Usiskin, “Square Patterns in the Pascal Triangle”

Back To Fibonacci Triangle Does the same identity hold in the Fibonacci Triangle?

No… But

)5)(21)(5()8)(8)(8(?

13)5)(21)(5()8)(8)(8(

Fibonacci Identity

Will this always hold?

|ace-bdf| =g iff

de

cgf

ba

1,1,1

,1,,1

1,11,

jiji

jijiji

jiji

cc

ccc

cc

)3(mod0 ij

iiiiii ffffff 23

322

21

3

iiiiii ffffff 23

322

21

3

Another Fibonacci Identity

121)()( 122

122

1 nnn

nnnn ffffff

fgh

ejia

dcb

121 ijacegbdfh n

Open Questions

What other conditions prohibit a sequence from corresponding to a row sum triangle and can all sequences that do not correspond to a row sum triangle be categorized?

What other ways can Fibonacci identities be obtained from the Fibonacci Triangle?

Open Questions

What is the ratio of the number of difference triangles created by using the numbers {0, 1, 2, … n-1, n} to the number of difference triangles created by using only the numbers {0, 1, 2, … n-1}?Note that a triangle randomly generated by just 0s

and 1s has 0 probability of being a difference triangle.

Thanks

I’d like to thank Professors Lengyel, Sundberg and Tollisen, as well as the rest of Occidental’s Mathematics Department.

Works Cited

Chen Chuan-Chong and Koh Khee-Meng, Principles and techniques on Combinatorics. World Scientific Publishing Company, September 1992.

V.E. Hoggatt and W. Hansell, The hidden hexagon squares, Fibonacci Quarterly, 9 (1971) 120.

A. F. Horadam, A Generalized Fibonacci Sequence, The American Mathematical Monthly, Vol. 68, No. 5. (May, 1961), pp. 455-459.

Kenneth Rosen, Discrete Mathematics and its Applications. McGraw-Hill; 5 Edition, April 22, 2003.

N.J.A. Sloane, Encyclopedia of Integer Sequences <http://www.research.att.com/~njas/sequences/>

Zalman Usiskin, Square Patterns in the Pascal Triangle, Mathematics Magazine, Vol. 46, No. 4. (Sep., 1973), pp. 203-208.

Eric W. Weisstein. MathWorld – A Wolfram Web Resource. <http://mathworld.wolfram.coml>