Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence

Theorem
There are an infinite number of rows of Pascal's triangle which contain $3$ integers in arithmetic sequence.

Proof
Suppose $\dbinom n k$, $\dbinom n {k + 1}$ and $\dbinom n {k + 2}$ are in an arithmetic sequence.

Then:

Since $n$ is rational, we require $8 k + 17$ to be a square.

Since $8 k + 17$ is odd, if $8 k + 17$ is square, then $\sqrt {8 k + 17}$ is odd.

Write $\sqrt {8 k + 17} = 2 x + 1$.

Notice that:

Using the substitution $k = \dfrac {x^2 + x - 4} 2$:

Each $x$ with $k = \dfrac {x^2 + x - 4} 2 > 0$ give a value for $n$.

Therefore there are an infinite number of rows of Pascal's triangle which contain $3$ integers in arithmetic sequence.

Also see

 * Rows in Pascal's Triangle containing Numbers in Geometric Sequence
 * Rows in Pascal's Triangle containing Numbers in Harmonic Sequence