Solving a Recursive Sequence in Sequence App
11-21-2014, 10:52 PM
Post: #7
 Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013
RE: Solving a Recursive Sequence in Sequence App
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
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 Messages In This Thread Solving a Recursive Sequence in Sequence App - maxlockett619 - 11-20-2014, 09:56 PM RE: Solving a Recursive Sequence in Sequence App - patrice - 11-20-2014, 10:40 PM RE: Solving a Recursive Sequence in Sequence App - Thomas Klemm - 11-20-2014, 11:33 PM RE: Solving a Recursive Sequence in Sequence App - maxlockett619 - 11-21-2014, 01:48 AM RE: Solving a Recursive Sequence in Sequence App - Joe Horn - 11-21-2014, 01:01 PM RE: Solving a Recursive Sequence in Sequence App - parisse - 11-21-2014, 06:49 AM RE: Solving a Recursive Sequence in Sequence App - Thomas Klemm - 11-21-2014 10:52 PM RE: Solving a Recursive Sequence in Sequence App - shaheer07 - 11-22-2014, 12:18 AM RE: Solving a Recursive Sequence in Sequence App - debrouxl - 11-22-2014, 02:41 PM

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