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Thursday, April 7, 2011

Pascal's Triangle

I just made a rather interesting discovery: in Pascal's triangle, the sum of the ith row is equal to the i-1th power of 2.
Row
Sum
Pascal's triangle:
1
1
0
1
0
0
0
0
0
0
0
0
0
0
2
2
0
1
1
0
0
0
0
0
0
0
0
0
3
4
0
1
2
1
0
0
0
0
0
0
0
0
4
8
0
1
3
3
1
0
0
0
0
0
0
0
5
16
0
1
4
6
4
1
0
0
0
0
0
0
6
32
0
1
5
10
10
5
1
0
0
0
0
0
7
64
0
1
6
15
20
15
6
1
0
0
0
0
8
128
0
1
7
21
35
35
21
7
1
0
0
0
9
256
0
1
8
28
56
70
56
28
8
1
0
0
10
512
0
1
9
36
84
126
126
84
36
9
1
0
11
1024
0
1
10
45
120
210
252
210
120
45
10
1
I was trying to remember how the construction of the triangle was done in LISP yesterday on the bus, and I decided this morning to try and figure out how to do it in Excel. Its extreamly easy in Excel, of course.


Added: ooooh, heres a fun question: if you made a "Pascale's Pyrimid" (ie, same idea but in three dimensions) what series would the sums of each layer make? Would it be the powers of three? If that turns out to be true, could it be generalized to n dimensions?

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