perplexus.info :: Just Math : Diagonal triangle sums

Diagonal triangle sums (Posted on 2004-03-15)

Consider the famous Pascal triangle, purposefully drawn in a somewhat lopsided way:


1
1  1
1  2  1
1  3  3  1
1  4  6  4  1
1  5 10 10  5  1
1  6 15 20 15  6  1
......................

Start at any number, and draw a line at 45 degrees, from bottom left to top right. (For example, if you chose the first "4" of the fifth row, the diagonal would also include a "1" and a "3")

How much do the numbers in such a line sum? Why? Can you prove it?

Submitted by Federico Kereki

Rating: 3.8571 (7 votes)

Solution: (Hide)

The Fibonacci numbers. To see why is this, consider two adjacent such lines, and apply Stieffel's theorem to produce the next line. In order to do so, if C(i,j) are the combinations of i elements taken j at a time, it will be useful to assume that, for all n, C(n,n+1)=0 and C(0,n)=0.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
A Different Presentation	Richard	2004-03-15 19:16:27
re: solution	e.g.	2004-03-15 14:22:39
solution	Charlie	2004-03-15 14:11:08
No Proof, though	nikki	2004-03-15 13:42:58

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