An Exploration Of Difference Triangles By Adrian Ferenc

An Exploration Of Difference Triangles By Adrian Ferenc April 3, 2008 Occidental College

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

Axiomatically Defined Structures n Algebraic Group n Equivalence Relation n Topology

Pascal’s Triangle

Pascal’s Triangle Continued

Difference Triangles n A triangle is a Difference Triangle if it adheres to the following condition: n For any n All entries are nonnegative integer values. .

An Example 36 51 15 72 21 6 102 30 9 3 145 43 13 4 1 19 2 429 132 42 14 5 2 1 1 90 9 1 297 28 3 0 Sloane’s A 059346 62 6 1 207

A Blank Triangle

Some Facts About Difference Triangles n Removing any number of the end columns still results in a difference triangle. n Multiplying our triangle by a “scalar” results in a difference triangle. n 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.

Increasing Triangles If in a difference Triangle, for all Or, equivalently, in any row m, Then this triangle is an Increasing triangle.

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

Facts About Increasing Triangles n n 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 .

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:

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,

So Why the Restriction? Let’s remove the absolute value restriction and see what happens… For any .

Let’s have our rows sum up to the numbers 1, 1, 2, 3, 5, etc…, i. e. , the Fibonacci numbers: Row 1 Rows 1 & 2 Rows 1 – 3 So there is no integral solution.

Example 1 n Let’s have our row sum sequence be 1, 1, 1, … Increasing n Unique n Multiply by k n

“Example 2” n What if we try the sequence 1, 2, 3, 4, … n For row 1 we get 1. n For row 2 we get x, y where x + y = 2, x - y = 1. n Clearly N

Nonexistent Row Sum Triangles n So for such a sequence to correspond to a row sum triangle. n Also n Similarly, our sequence cannot be infinitely decreasing.

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

Example 2 n Let’s try the row sum sequence 1, 3, 5, 7, … n Not an increasing triangle. Unique? n

Non-Uniqueness

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

Back To Pascal’s Triangle Note that here the rows sum up to 1, 2, 4, 8, … or

Example 3 n Let’s have our sequence be 2, 4, 8, 16, …

Example 4 n Let’s have our sequence be 1, 3, 9, 27, …

Example 5, A Generalization n So we’ve seen and . n Are we able to generalize this to ?

Yes! n In fact we can do more! n We can create a row sum triangle whose rows sum to:

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

“Proof” Outline of proof: n Part 1: Show n Part 2: Show that this Triangle is increasing by showing that ¨ This tells us that

“Proof” Continued n Part 3: Here we show that: n n Combining parts 1 – 3 we get:

Taste Of Proof

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

The Fibonacci Triangle

Fibonacci Triangle Continued n Is it unique? No Since

Back To Pascal’s Triangle Zalman Usiskin, “Square Patterns in the Pascal Triangle”

Back To Fibonacci Triangle n Does the same identity hold in the Fibonacci Triangle? n No… But

Fibonacci Identity n Will this always hold? n |ace-bdf| =g iff

Another Fibonacci Identity

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

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

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

Works Cited n Chen Chuan-Chong and Koh Khee-Meng, Principles and techniques on Combinatorics. World Scientific Publishing Company, September 1992. n V. E. Hoggatt and W. Hansell, The hidden hexagon squares, Fibonacci Quarterly, 9 (1971) 120. n A. F. Horadam, A Generalized Fibonacci Sequence, The American Mathematical Monthly, Vol. 68, No. 5. (May, 1961), pp. 455 -459. n Kenneth Rosen, Discrete Mathematics and its Applications. Mc. Graw-Hill; 5 Edition, April 22, 2003. n N. J. A. Sloane, Encyclopedia of Integer Sequences <http: //www. research. att. com/~njas/sequences/> n Zalman Usiskin, Square Patterns in the Pascal Triangle, Mathematics Magazine, Vol. 46, No. 4. (Sep. , 1973), pp. 203 -208. n Eric W. Weisstein. Math. World – A Wolfram Web Resource. <http: // mathworld. wolfram. coml>