## PART 1: sequences, patterns, and discrete calculus

- Problem: can you find the formula for a sequence?
- Try discrete differences

- Problem: that’s the pattern, but what’s the formula?
- Write it in terms of antidifferences
- Antidifferences are indefinite sums up to a constant
- Write it in terms of sums

- Problem: that’s the sum formula, but what’s the polynomial formula?
- Summing sequences sums their differences
- Multiplying a sequence by a constant multiplies its difference by a constant
- Use the geometry of rectangles to solve for the geometry of a triangle

- Problem: that was hard, is there an easier way to find a more
general polynomial formula?
- List off the difference for each monomial
- Rearrange the chart in terms of sums
- Solve for each sum
- Use the chart as a reference

- Problem: can you find the level of an (easy) sequence?
- (introduce without level, then introduce level)
- Take differences until it’s constant

- Problem: can you find the level of a (medium) sequence?
- Differences don’t work
- Sums and differences cancel out
- Sums are like negative differences
- Take sums till it’s constant, and make that number negative

- Problem: finding the level of a hard sequence
- Differences don’t work, neither do sums
- The answer is , but getting there is complicated, save it for part 2

## PART 2: recurrence transformations, weight analysis, and functional roots

- Problem: what exactly is a sequence’s level?
- Put simply, the level of the sequence the degree of its polynomial expression
- Some functions, which aren’t polynomials, have no level

- Problem: what about sequences which have a degree but aren’t
polynomials?
- The polynomial definition works for polynomials, but clearly that’s only part of a larger phenomenon
- The degree is how many sums/differences you need to apply to a constant to get a function

- Problem: ok, but what exactly are sums and differences?
- Mess with variations on the difference definition to get a more general “recurrence transformation” phenomonon
- Reframe the sum as a recurrence transformation
- Use weight analysis to define higher order sums/differences as recurrence transformations

- Problem: what does this tell us about level?
- Transforming two sequences adds their level
- You can find level by splitting into factors
- Level is just how many unit factors the sequence can be split into

- Problem: why is the sum the unit?
- Easy answer: it lines up nicely with polynomials, but that’s not very satisfying
- Try starting with a different base unit and base inverse
- Try finding the level of some sequences
- Everything still works, just scaled differently
- Interestingly, some sequences can now have non-integer level
- It actually is kind of arbitrary

- Problem: could we have found the sum sequence from the double sum
unit sequence?
- Easy answer: use the difference, but that’s kind of cheating
- Use a table to visualize transforms
- Reverse the problem to solve for each item on the sides of the table
- Vocab: functional root

- Problem: can we find the functional root of the sum sequence?
- Use the table method from before
- There’s no easy pattern, but it works!

- Problem: now can we return to the level of the hard sequence?
- Use our rules and some algebra to work out that the new sequence has level
- Use our rules to figure out that the sequence from before has level

- Problem: start with a level and find a sequence
?
- Use the functional root technique
- Show that it works for any dyadic rational level

- Problem: can you find sequences with more numbers
(
)?
- Functional square roots won’t help us, since is non-dyadic
- Functional cube roots could work for , but calculating them is a hot mess
- Functional roots are basically hopeless for
- No clue where to even start with
- This
*is*possible, but getting there is complicated, save it for part 3

## PART 3: convolutions, derivatives and power series

- Problem: can we find a pattern for the weights in the
level
sequence?
- Trying difference techniques won’t work
- Try on just the numerator and denominator, that still doesn’t work
- Try and reframe the problem as shifted and scaled copies
- Notice location addition and value multiplication
- Tangent to polynomials
- Reframe polynomial multiplication as shifted and scaled copies
- Notice location addition and value multiplication
- Show equivalence between functional root problem and the polynomial square root problem
- Demonstrate that the equivalence works by testing it on the second order sum sequence

- Problem: ok, but how do you find a pattern?
- Use algebra to show
- Reframe the problem as power representation of
- Show that our manual calculations for the sequence work to approximate it

- Problem: how does that help us? where’s the pattern?
- Define an analytic function, and show that is one
- Take: calculus says that every analytic function has a power series
- Take: the coefficients will be value, rate, rate of rate, etc
- Take: you can calculate rate by taking a derivative
- Take: you can find the derivative of any analytic function using these rules
- Demonstrate that evaluating derivatives of produces values of the sequence

- Problem: ok, but how do we get from there to an actual, closed form
pattern?
- Unpack the derivative rules to show patterns in the functions
- Simplify the patterns for the special case
- Derive a real pattern in closed form
- Review: to find a level sequence, we took the power series of and derived a pattern from the differentiation rules

- Problem: can we return to find sequences with weird weights?
- For , just take the series coefficients for
- Test that it genuinely does work out to the functional cube root of
- For , just take the series coefficients for
- You can’t test it the same way, but it does work

- Problem: what about the level i sequence?
- Complex exponentiation works with new rules
- We don’t need them though, we can just plug and chug
- Deriving a pattern
- Finding the sequence
- Demonstrating that it works

## Outro

- Credits:
- Mythologer’s video
- Supware’s video
- Morphocular’s video

- Unanswered questions:
- What actually is complex exponentiation?
- What other sequences have no level?
- What if we start with a level-less unit transformation?

- For next time:
- Can we leverage the connection between differences and derivatives to extend our techniques to the continuous relm?