Odd numbers in Pascal’s triangle

Here’s an interesting theorem I ran across recently.

The number of odd integers in the nth row of Pascal’s triangle equals 2b where b is the number of 1’s in the binary representation of n.

Here are the first few rows of Pascal’s triangle:

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
1  7  21 35 35 21 7  1
...

We count rows starting from 0, so for n at least 1, the nth row has n in the second column.

Notice, for example, that on the 6th four of the entries are odd: 1, 15, 15, 1. The binary representation of 6 is 110, so b = 2, and 2² = 4. In the 7th row, all eight entries are odd. The binary representation of 7 is 111, and 2³ = 8.

There are a couple quick corollaries to the theorem above. First, the number of odd numbers in the nth row of Pascal’s triangle is always a power of 2. Second, in row 2k-1 – 1, all entries are odd.

This post is a slightly expanded version of a Twitter thread I posted on @AlgebraFact this weekend.

More on Pascal’s triangle

One thought on “Odd numbers in Pascal’s triangle

  1. Pascal’s triangle has many interesting features. For example it is not known whether a number exists that appears in it more than 8 times, or whether any number other than 3003 appears exactly 8 times. It is also known there are infinite numbers appearing exactly 6 times, the smallest one being 61 218 182 743 304 701 891 431 482 520

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