Solving a Recursive Sequence in Sequence App

[Thomas Klemm]

Quote:(…) where you can input a sequence and be given a rule back?

For polynomial sequences you can apply the forward difference operator \(\Delta\) consecutively until you get just 0s.
For the tetrahedral numbers we get:
\[
\begin{matrix}
U &: & {\color{Red} 0} & & 1 & & 4 & & 10 & & 20 & & 35 & & 56 & & \cdots \\
\Delta U &: & & {\color{Red} 1} & & 3 & & 6 & & 10 & & 15 & & 21 & & \cdots & \\
\Delta^2 U &: & & & {\color{Red} 2} & & 3 & & 4 & & 5 & & 6 & & \cdots & & \\
\Delta^3 U &: & & & & {\color{Red} 1} & & 1 & & 1 & & 1 & & \cdots & & & \\
\Delta^4 U &: & & & & & {\color{Red} 0} & & 0 & & 0 & & \cdots & & & & \\
\cdots &: & & & & & & & \cdots
\end{matrix}
\]

Then use Newton's forward difference formula to write the sequence as a sum of binomial coefficients:

\(U(n)=0\binom{n}{0}+1\binom{n}{1}+2\binom{n}{2}+1\binom{n}{3}\)

Cheers
Thomas