C4. Embedded sequences
This post continues from where the previous one left off. As an obligatory disclaimer, if you haven't read the previous parts I highly recommend that you do before continuing with this one. Each post in this series builds on top of the earlier ones making them difficult to grasp in isolation unless you are already familiar with the subject matter. In the previous part we looked a bit into the coefficient sequences arising from the geometric power progressions of the various proportionality constants which are inextricably linked to the continued fraction gap patterns investigated in the first part of this series. The constants can also be derived separately in the context of so-called optimal sections — the well-known golden ratio being the first non-trivial case.
In this part we will first see how the coefficient sequences of the inverse powers (also discussed in the previous post) relate to the coefficient sequences of the positive power progressions. We will find some similarities, but the differences are equally interesting. Lastly, we will shift our perspective slightly to uncover a neat connection with something rather unexpected. Again, I would like to offer a continued fraction fractal to serve as a mood backdrop before getting down to the nitty-gritty. In the image, you can see a string of different sized dots which – under recursive transformation – weaves itself into a strange self-similar web of pearls. The dots are equivalent to the fraction gaps which are more readily visible in another image. Half of the entire gap structure is determined by the proportionality constants of the different optimal sections.
Inverse sequences
Let's again start simple, with the golden ratio. Like in the previous part, as we painstakingly expand and evaluate and simplify the products we are again confronted with the very same Fibonacci sequence we saw in the forward geometric progressions of the golden ratio. Except this time there is an alternating sign pattern interwoven into it, similar to the ones we saw in the inverse factorizations themselves.
Click to see the full beauty
This is quite literally only the half of it. These signed Fibonacci numbers are also what you get if you extend the more conventional Fibonacci sequence to negative indices, meaning that the same recurrence relation governs both sequences and that they can be joined together into a fully symmetric whole whose apparent direction of "motion" in terms of indexing is from negative to positive. Although, due to its simple nature the recurrence relation is perfectly reversible so this direction is determined solely by established convention. This symmetry of the positive and negative coefficient sequences is a well-known property of the golden ratio. The full sequence as well as the three different, but equivalent forms of the underlying recurrence relation are shown below.
If you ever find yourself stranded on a Fibonacci sequence, these recurrence relations will help you move around and get you to where you're going (provided its on the sequence). At this point one might be tempted to extrapolate that all coefficient sequences are similarly symmetric and that the forward and backward geometric progressions are always joined neatly at the middle. Such an extrapolation would turn out to be wrong though. Naturally, since we could not find factorizable inverses for some of the constants we cannot produce the backward progressions for them either — in some sense they escape outside our otherwise neatly self-enclosed proportional number systems. Since there's no such possibility in the case of the forward sequences (due to the maximal optimality of our sections), some of them are bound to be left without a symmetric counterpart. However, this is not the only confounding factor we encounter if we attempt to extrapolate. Consider the backward (or inverse) sequences of ρ and σ.
Instead of the forward and backward sequences of individual constants being simple reflections of themselves (with different sign patterns) the backward ρ-sequence is actually a forward σ-sequence with alternating signs, and similarly, the backward σ-sequence is a sign alternating twin of the forward ρ-sequence. This seems a bit odd when we recall the kinds of recurrence relations that governed the forward sequences — ρ's recurrences were an interleaved pair of two-term sums, while σ had one three-term and one two-term sum recurrence. So does the recurrence pattern just abruptly change mid-sequence in both cases? Otherwise it would seem too convenient that the reversals of two different recurrence patterns just somehow ended up corresponding perfectly with the forward sequences of each other, but it turns out that this is indeed the case! As we extend the forward ρ-sequence to the direction of its negative indices it reproduces the sign alternated forward coefficients of σ, and vice versa for σ. It would seem highly counter-intuitive that this could happen with recurrence relations of different flavors, ie. three-term vs. two-term recurrences. This correlation becomes even stranger for higher optimal sections.
Up until this point we have been implicitly working according to the same term ordering rules we used to build the two initial sets of forward sequences, namely that we can figure out the ordering simply by first glueing together the directly correlated coefficients and then filling in the middle with whatever is left. Ordering things in this way has resulted in a simple reversal of the forward term orders: the backwards ρ-sequence uses a (ρ, σ, 1)-order which is the simple reverse of the forward (1, σ, ρ)-order, and similarly for σ. Since this has the effect of making the sequence symmetries undeniable, term-order reversal seems like the right way. For now.
Symmetry breaks
Continuing in a similar manner with the backward sequences of the next optimal section yields similar results albeit with the absence of the β inverse — the backwards α-sequence corresponds to the forward γ-sequence and backward γ-sequence corresponds to the forward α-sequence. Which is now even more striking considering the latter is defined in terms of one four-term, one three-term, and one two-term recurrence relations while the former uses three two-term recurrences! As a minor side-note: unlike for the forward sequences, here (as well as earlier) I've chosen to keep the zero coefficients around so as not to obscure the lockstep pattern of the alternating signs.
Unfortunately, this is where our free ride ends as far as the term orderings are concerned. For the next optimal section a simple reversal of forward order no longer automatically guarantees matching coefficient sequences. However, the reverse ordering does still continue to work for the first and last sequences and these anchors of regularity also remain each other's coefficient-wise counterparts as far as I'm able to verify. Sadly, I haven't been able to figure out a neat derivation for the inverse term orderings of the sequences inbetween — their apparent chaotic nature refuses to yield its secrets. There is one way to programmatically generate orderings that works for some backward sequences, but I'll leave that for later since explaining why it works at all is difficult without first involving Pafnuty Chebyshev. Luckily enough, brute force searching as a fallback is made possible by the fact that the backward sequences will always correspond to at least one of the related forward sequences — the only truly unique sequences being those which cannot be reversed (like β). So, as long as we can find the forward counterpart (with some light set analysis) we can always figure out the necessary term order that produces a coefficient sequence match. It's not pretty, but it works.
With the application of some brute force we can now line up the θκλμ-sequences of the next optimal section which allows us to notice a curious extended symmetry in their correspondences. In addition to the smallest-largest correspondence we already discussed, now the forward sequence of the second smallest proportionality constant seems to correspond with the backward sequence of the second largest constant and the forward sequence of the latter corresponds with the backward sequences of the former. They form a new symmetric sequence-pair similar to the one formed by the sequences of the smallest and largest constant — θ and μ. But again, jumping to any conclusions would be premature since as soon as this symmetry appears it also breaks. Another thing that we seem to lose for the κ and λ sequences is the perfectly alternating sign patterns that we have so far been enjoying.
In the formulations above the signs of the backward sequences have been stripped away by the means of absolute values in order to satisfy the equivalences. In case the n-dependent exponentation of -1 seems strange to you, it is a common way to introduce alternating sign patterns into mathematical sequences — with even exponents the term evaluates to one and with odd exponents the term evaluates to minus one. In effect, it replicates the perfectly alternating sign pattern we saw earlier and as the equations indicate this sign pattern still applies to the first and last sequences. Serving again as the exceptions, they seem to maintain this sign pattern in the case of all optimal sections.
The middle sequences are another story. Although their sign patterns are still balanced in the sense that there is an equal number of positive and negative coefficients, the distributions of the different signs have become much less straightforward. Naturally the sign sequences depend on the ordering we choose, but unless we detach our sequences from their originating coefficients entirely it is not possible to form perfectly alternating patterns. Only by interleaving the coefficients of different powers of κ and λ could we hope to recover the perfect sign alternation and later on even that won't work. So for now, the coefficient-wise matches seem a far stronger signal than the sign patterns and so we continue to rely on our brute force ordering despite the disruption.
As mentioned earlier, the trivial reversed-order symmetry of the sequences breaks for the next optimal section. Note the slight simplification in the subscript notation in order to reduce the symbolical clutter that is starting to overtake us. Although the first and last sequences still maintain their reciprocality and alternating sign pattern, the forward and backward sequences of the other constants have become somewhat confused. Now the middle constants Ψ₃ and Ψ₄ form a sequence pair while the leftover sequence of Ψ₅ serves as its own counterpart — just like what happened with the Fibonacci sequence. As we keep producing more inverse sequences and matching them with forward sequences (always successfully) these jumbled connections become even worse and seem to offer no clear structure for us to investigate.
Really the only thing that seems to hold true for the middle sequences is that if the forward sequence of A corresponds with the backward sequence of B, then the forward sequence of B also corresponds with the backward sequence of A. In other words, the sequences always form pairs instead of something more complicated where three or more sequences would be interwoven together. Occasionally a sequence will form a symmetric pair with itself making A equal to B — as is the case with Ψ₆,₅ and Ψ₈,₄ in the figure above.
The accompanying sign sequences are also mostly an indecipherable mess and many do not even stay sign-balanced, as some sequences simply contain fewer positive coefficients than negative ones (or vice versa). The sign sequences do appear to always be periodic, but besides that there isn't all that much to go on. You can see some of the non-trivial sign sequences produced by the brute forced orderings below.
Click for more
The periodicity of the sign sequences seems to be linearly dependent on the degree of the optimal section (p=2N-2) and the sign patterns tend to be balanced over these periods, but not necessarily — as evidenced by some of the patterns above. Although we have managed to learn that the inverse sequences (when available) can always be made to correspond with the coefficients of one of the associated forward sequences, we seem to have no way of moving forward with our current direction of investigation. So before we drown in the chaotic waters of the inverse sequences it might be best to shift our perspective slightly and attempt to approach the problem from a different direction.
Reversable recurrences
Let's forget the forced coefficient matching for a while and instead concentrate on the recurrence relations that govern the sequences. We've already formulated some of these relations in the previous post as well as in this one, but mostly their treatment has been superficial. We would also like to ditch the standard recurrence notation since it gets really cumbersome really fast. Instead, we will resymbolize the F(n-x) terms with capital letters to make them a bit less distracting. The vertical number columns below are meant to represent the extended sequences of the proportionality constants ρ and σ. The bidirectional recurrence relations in terms of alphabetical symbols are the re-expressions of the more conventional rules found in the previous post.
The primary (or forward) recurrence relations are shown on the bottom (corresponding to D and E), while the reversals derived from these forward recurrences are shown on top (corresponding to A and B). Since both chains are governed by two interleaved recurrences the rulesets cannot be positioned or moved about arbitrarily, and whether sliding the ruleset backwards or forwards the movement must always occur in steps of two to avoid mixing the recurrences. For example, suppose we didn't know any of the number below E. Simply by sliding our coordinates and their rules downward by two steps (so that C becomes the new A) we would then be able to calculate the two missing values positioned at the new locations of D and E. Inching our way forward two steps at a time we can keep producing new numbers for as long as our patience allows. It should also be pretty easy to see that when moving forwards in the sequence we only need to know the values of A, B, and C after which the rest will follow. Furthermore, since we are only ever moving two steps at a time we will never run into a situation where one of these base values would be unknown. The backward motion works in a perfectly symmetric manner, again two steps at a time.
The ruleset simplification made for σ (shown in parentheses) should give you small hint as to why the sequences form a forward-backward symmetry pair. We already know that the overall shape of the recurrence relations can be seen from the associated multiplication tables, and with a little effort we can work out that despite their initial outlook the recurrences of the last sequence can always be simplified into a number of two-term relations. In this simplified form it no longer seems quite as surprising that the first and last sequences should continue to correspond so neatly with each other. The recurrence symmetry becomes all the more obvious if we reverse the alphabetical labeling of F(σ). The strictly linear recurrence sequences also provide a straightforward explanation for why a simple reversal of the forward term order should be the desired inverse term order. At least in the simple cases of the first and last sequence. But what about the not-so-nice sequences?
The number of interleaved recurrences has increased by one, so this time our mathemolecular machine must advance in steps of three — when moving forward D always becomes the new A, and when moving backwards D becomes the new G. The forward direction works just like before, but as you might have guessed the backwards direction of β presents some problems. Suppose that we knew none of the numbers above A, and when we slide our ruleset upwards in an attempt to produce those missing numbers we find we are unable to do so! Since C is defined to be equal to E minus B we cannot calculate its value since B is also unknown, and calculating the value of B would require us to know the value of C! The only way to break this deadlock is to make a guess. While it is still possible to guess right for the first three numbers above the initial position of A, after that it becomes impossible (as indicated by the hyphens). Since the recurrences are analogous with the related inverse factorizations, it is perhaps unsurprising that attempting to backpedal further with the recurrences leads to contradictions similar to what we saw when attempting to produce factorization for the inverse of β.
This connection between irreversible recurrences and unfactorizable inverses holds true also for the other asymmetric sequences. So with reasonable confidence we can conclude that just as the forward recurrence relations are tied to the multiplicative factorizations, so are the reversed recurrences correlated with the inverse factorizations. Indeed, the highly factorizable nature of the optimal section products and inverses makes them behave as if they were recurrence relations — the two different representations are really just two sides of the same coin. For rather more elusive reasons the irreversibility of some of these recurrences is linked to a seemingly unconnected factorizability pattern of a different kind (as seen in the previous post). However, in itself the reversibility of recurrences does not guarantee smooth operations and some sequences seem to (at least superficially) rely on value "look-ahead" where the value of an unknown relies on other unsolved unknowns.
Here, even after some suitable simplifications the reversed λ-sequence seems to run into a similar deadlock as we saw with the β-sequence — deriving the value of D without relying on the values of A, B, or C seems impossible. However, since we already know that the λ-sequence must be reversible we can safely conclude that apparent deadlocks do not necessarily result in irreversibility. Only once the recurrence relations produce an unavoidable contradiction does the associated sequence become irredeemably irreversible. Indeed, with some applied ingenuity and effort we can find a recurrence relation for D which does not rely on look-ahead, although it does differ from the others in some respects. Firstly, the relation D=F+G-H can only be derived by a sequence of substitutions that is entirely unnecessary in the case of the other simpler recurrences. Secondly, its contributing terms deviate from the otherwise uniform signage pattern. This difficulty is a hallmark of the misbehaving middle sequences.
In the context of recurrence relations the bidirectional symmetry of the λ and κ sequences is somewhat harder to see since as mentioned earlier, they are the first sequences for which a simple term order reversal – forced upon us by the linear recurrence rules – fails to produce a coefficient-wise match. It almost feels like all of the difficulties plaguing the backward sequences are somehow connected. The problems of reversibility, of tangled up symmetry pairs, and of brute forced term orderings appear to be under the influence of chaos. For some reason this chaos is entirely absent in the forward context.
So unfortunately, at this point in time my best explanation for the patterns found in the inverse factorizations and their associated coefficient sequences is: it's really complicated. Which leaves us with a wealth of unsolved problems. What gives rise to the symmetry pairing chaos of the middle sequences? Why should the sequences be self/pair-symmetric at all? Why are the necessary term orders so convoluted when the symmetry pairs are so perfect? Is there some regularity to these orderings? Does a simple construction pattern exist for the signs of the backwards sequences? These mysteries shall have to wait as we dive into another rabbit hole.
Power structures
Instead, let's see if we can fit our abundance of sequences within a bigger context. In the previous post we found that as we increased the degree of our optimal section the associated coefficient sequences appeared to stabilize out of initial "noise". In this context, the Fibonacci sequence seemed like an abbreviation of something more grandiose, although none of the stabilized sequences seemed to carry any weight as far as the Online Encyclopedia of Integer Sequences was concerned. Back when I was originally looking into these crystallized sequences I managed to stumble – quite unintentionally – upon a representation that offered the bigger context to me as if on a silver platter. At first, it may seem like an unwarranted leap, but I promise that the ends will justify the means.
Let's start by looking at the sequence that seems to stabilize the fastest — the sequence associated with the first proportionality constant of each optimal section. Each sequence in this infinite family of sequences is quantitatively different, but qualitatively the same. The Fibonacci sequence may look nothing like the ρ-sequence, but the underlying binary recurrence is the same, and as we move up the ladder of optimal sections the prefix resemblances become undeniable. With the stabilization come the zero coefficients which, for higher optimal sections, make up most of the sequence in the beginning. As we extrapolate our way to the apeirogonal infinisection the sequence is practically all zero with only intermittent exceptions. In fact, it is precisely the abundance of zeroes that makes the stabilization possible. Although the infinisectional F₂ sequence is defined as the rather ridiculous interleaving of an infinite number of two-term recurrences, the mostly zero starting state manages to make the vast majority of these recurrences inert. Amidst this sea of zeroes the finite starting state evolves into a succession of miniature sequences, embedded within the larger F₂.
In our previous brief encounter with the infinite-sided apeirogon we saw how the number 2 can be considered to be the end of the proportionality constant sequence that begins with the golden ratio and continues with ρ, α, and θ — it is the largest of the smallest proportionality constants (ignoring the unit constant one). This means that the extrapolated F₂ sequence above consists of coefficients that make up a powers-of-two geometric progression, although this somewhat non-obvious due to the unconventional way the products are factorized. Our deduplication of the base coefficient pairs which we adopted to make the sequences also serves to conceal the powers. Let's decompose the miniature sequences into their exponential parts and then stack the coefficients based on their associated multipliers (in ascending order). This is the part that might seem like a leap.
Once our hard-won coefficient sequence scaffoldings fall away we are left with a neat little number triangle. In the interest of clarity all the zeroes have been left out from it because with them included the non-zero coefficients would be lost in the noise. There is also the small matter of the infinite number of zeroes left out from the right side of our table, but for now we shall not trouble ourselves with these minor details.
The leftmost column in our table of numbers indicates the power of two equivalent of the row factorization while the topmost row associates each column of coefficients with a particular proportionality constant term — recall that the proportionality constants of the apeirogonal infinisection are simply the natural numbers. In this representation the coefficients which were previously deduplicated now make up the near identical n and 2n columns. The miniature sub-sequences of F₂ are cut in half at these deduplication points, with the leading part placed onto an odd-exponent row (right to left) and with the trailing part flowing onto the following even-exponent row (left to right). As an example of use, the fifth power of two can be factorized as (5x2) + (4x4) + (1x6) according to our triangle.
From our new perspective the previously obscured nature of the stabilized coefficient sequence starts to become apparent, and suddenly our humble collection of numbers has gone from "sorry, no matches" to "probably the longest entry in the OEIS". When read from top to bottom the first coefficient column comprises the so-called Catalan numbers which relate to an enormous number of different combinatorial problems, and in its entirety our humble number triangle is actually equivalent to Catalan's triangle. Although not quite exact copies — the columns and rows of our triangle are Catalan's diagonals (and vice versa). Also, the zeroes in our triangle or rather the holes left by their omission have no obvious place in Catalan's context. Visually the most pleasing combinatorial problem relating to the Catalan numbers is perhaps counting the non-crossing partitions of a set, or as below, counting the non-crossing partitions of a pentagon.
Figure 2. Non-crossing partitions of a 5-set
There are a total of forty-two such non-crossing partitions including the empty partition at the top and its polar opposite at the bottom. The number forty-two is the fifth Catalan number and also according to our triangle the number of ones in the somewhat curious factorization of the tenth power of two. Due to the reintroduced duplication forty-two is also the number of twos in the equally strange factorization of the ninth power of two. Above them both sits the number fourteen which is the fourth Catalan number — the total number of non-crossing partitions of a four element set (or a square) and the number of ones in the factorization of 2⁸. You can probably see the relation. The generalized Catalan numbers found both in our triangle and in Eugène Catalan's triangle relate to a slightly more complicated combinatorial problem which funnily enough is connected to the recurrence relations of our original sequence.
Having arranged our coefficients into this neat triangular pattern has made their binary recurrence relations almost trivial. The somewhat awkwardly patterned infinite interleavings have been replaced by perfect uniformity and every number in the triangle is simply the sum of two numbers above it — one to its top-left and one to its top-right. This strict diagonality of the recurrences also explains how the hidden zeroes can remain so undisturbed in the liminal spaces left by the non-zero coefficients. They are perfectly disjoint from all the Catalan business and simply go on with their null lives right next to a huge combinatorial explosion. When drawn out, the diagonal recurrences form a sort of lattice structure which conveniently offers us the backdrop for the generalized Catalan numbers.
Imagine you suddenly found yourself standing at the very top of this mathematical lattice and despite (or due to) the strange circumstances decided to go for a stroll. Moving about in this fictional lattice-land has its own strict rules: you are only allowed to travel along the lattice, a single connecting diagonal at a time, and you can only move downwards. So really, no matter where you find yourself standing you only have the option of two directions. Following these rules, how many unique paths are there that go from the top-left corner of our lattice to some other numerical location on it?
Figure 3. Lattice paths from C₀ to C₃
As you might have guessed, the generalized Catalan numbers are the answer to this very question. Each number in the triangle is equal to the number of unique downward paths leading to it from the top-left corner. In the diagram above you can see the five different paths that can be formed between the nodes corresponding to the first and fourth Catalan number. When tilting your head to the right the lattice paths take on the shape of neat little mountain ranges. If you like the scenery, here's the set of fourteen unique mountains associated with the fifth Catalan number. Since there is only one path connecting the two bottom-most nodes of our truncated lattice their values will always end up being the same — thus the duplication of the Catalan numbers in the two first columns of the triangle.
Similar enumerations can be produced for all the other node-numbers in our lattice arrangement. It should be fairly easy to see why the numbers on the rightmost edge of our number triangle are all ones, as together they form a completely singular path along that diagonal. Although this lattice-walk problem may seem artificial or contrived, it is quite simply a different representation of the same underlying mechanism also described by the recurrence relations. As mentioned earlier there's a very large number of such different combinatorial interpretations one of which neatly relates (at least somewhat) to the sign sequences we looked into earlier. We can reinterpret the two different diagonal directions to be the two signs, with the rows indicating the total number of signs in the sequence, and with the vertical columns telling us the sum of the sign-sequence paths. Balanced pairs of positive and negative cancel out, therefore leading all fully balanced sequences back to the originating column (shown in purple below).
Figure 4. Combinatorial sign-paths
Note that if we adhere to the lattice-travel limitations set out earlier the associated sign sequences will never end up net-negative. Of the unbalanced sign sequences which end up net-positive, the green paths have one extra plus, red paths have two, and so on for each distinct vertical column of lattice nodes. Combining these twelve sign sequences with their sign-reverse counterparts gives us the set of all possible sign sequences with a maximum length of four. Although the truncated size of our example offers us few examples, notice how the sign-paths also have a kind of additive and multiplicative structure of their own. The additive property should be pretty obvious — joining two paths is equivalent to simply concatenating two sequences. Multiplying a sign sequence by two is the same as duplicating each sign in the sequence, resulting in the shape of the path becoming twice as large. But this is quite enough about these lattices. The last thing I want to mention about the Catalan numbers is that they also make an appearance in the context of the Mandelbrot set!
Reminiscent of the miniature sub-sequences that stabilized out of the initial noise of the proportionality constant coefficient sequences, the Mandelbrot coefficients stabilize one by one as the sequence is recursively iterated. This linear stabilization is obviously vastly outpaced by the exponential growth of the full polynomial form (truncated above with three dots). Despite this disparity, if we iterated the Mandelbrot function an infinite number of times its coefficients would become equal to the Catalan numbers. Which feels very counterintuitive and seems like it just couldn't be true, but that's the conclusion offered by Donald Cross. There's a lot more that could be said about Catalan's numbers and his triangle, but since in our context they apply only the F₂ sequence it is time we wrapped up this rabbit hole. So what about F₃ and friends?
Riordan and quantum mechanics?
So let's see what the same treatment does to the other F-sequences. We already know that they also end up stabilizing into a bunch of nested sub-sequences which can naturally be stacked into a number triangle of their own. While the Catalan triangle contained the factorizations of powers of two, the number triangle constructed from the F₃ sequence corresponds to the factorizations of powers of three. This pattern continues in a similar fashion for the other stabilized F-sequences. Interestingly there is one obvious difference between the triangles corresponding to powers of even numbers (Catalan's triangle) and odd numbers (see below).
Our way of factorizing the powers of three leaves all the columns corresponding to even terms empty, so for reasons of simplicity they are omitted in their entirety along with all the zeroes. The interleaving pattern we saw in the Catalan's triangle which managed to intertwine all the term-columns with each other is now gone and we are left with an alternating pattern of solid columns. The only exception to this solidity is a single hole left by a zero in the factorization of three to the power of one (or just three). The rightmost diagonal in our new number triangle is still all ones, the diagonal below it contains a slightly offset linear sequence of the natural numbers and again, the two leftmost columns are offset duplicates of each other. When read from top to bottom these primary columns give us the Riordan numbers which are not quite as well-known or as widely applicable as the Catalan numbers, but still important in their own right.
The triangle also obeys a three-term (or ternary) recurrence relation — each number is the sum of the three closest numbers above it, for example 15+15+10 = 40. Unfortunately we run into a slight complication as we try to apply this recurrence to our leftmost column as clearly 36+91 is not equal to 91. In the case of Catalan's triangle we were able to gloss over the mirrored negative half, but this time the increased width of our recurrence relation forces us to consider it. Including this negative half in our considerations allows the ternary recurrence relation to function without exceptions. Although then one might be tempted to ask: where do the initial ones come from? Or in other words, what's above the balanced triangle? In the case of the n-column this sign-symmetry conspires to cancel out the vertical numbers so that -36+36+91 does become equal to just 91. As hinted at by the three-way recurrence relation, the lattice structure associated with the Riordan triangle is slightly more complicated than before.
Figure 5. Paths along a 3-lattice
As expected the 3-lattice now allows for three possible downward directions, although in the case of the n-column the destructive interference of the adjacent negative column forbids us from using the vertical paths within that column. So all paths that end up on the origin column must do so from the right. The diagram above showcases the fifteen different paths that can be formed between the lattice nodes corresponding to the first and sixth Riordan numbers. Since the second vertical node is unreachable due to our self-imposed traveling restrictions the lattice ends up looking a bit lopsided.
The further F-sequences continue the trends already established by the first two non-trivial sequences (remember that F₁ is all ones) — F₄ gives us the factorizations of the powers of four, with the associated number triangle defined in terms of 4-way recurrence relations, and thus being equivalent to a 4-lattice with an appropriate degree of directional freedom. As mentioned before, the columnar structure of the resulting number triangles follows an alternating pattern. They also start to become rather unwieldy due to their quickly increasing horizontal girth. They also stop following a strictly diagonal growth pattern, and instead the right edge of the triangles becomes terraced. Another deviation is seen in placement of the primary column duplicates. It is a bit hard to make out with the small number of rows shown in the linked image, but the duplicate column moves one step to the right with each new F-sequence. This pattern was obscured in the case of the Riordan triangle by our omission of the zero columns. These primary sequences are shown below.
The reason why the even-index sequences appear to grow faster is because of the interleaved zeroes that are omitted — we are effectively showing only every other number for the even sequences while the odd sequences contain all their numbers — correcting for this disparity makes the index-dependent growth rate more obvious. As mentioned earlier and as indicated by the notation the Catalan numbers can be found on the first row, and below them the Riordan numbers. Unfortunately, as an indication of increasing obscurity that is where the personified sequence names run out.
Fortunately however, the Online Encyclopedia of Integer Sequences still knows of them. The numbers in the D₄ sequence are the "degeneracies of entanglement witness eigenstates for spin 3/2 particles" or simply A264607. The numbers of D₅ have an equally cryptic description, being the "number of noncommutative SL(2,C)-invariants of degree n in 5 variables" (A007043). Rest of the sequences follow the terminology of D₄ and correspond to various degeneracies of entanglement witness eigenstates for different spin configurations (A272391, A264608, A272392, and A272393).
Most of these sequences are referenced by just two research papers in OEIS. One is titled "From Entanglement Witness to Generalized Catalan numbers" and the other one is simply "Spin Multiplicities". I interpret the sparsity of references to mean that these sequnces are mostly just very niche research curiosities. Somewhat more notably though, the special linear group of SL(2,C) happens to be related to the special unitary group SU(2) which is used to model one of the four known fundamental forces of our universe — namely the weak interaction.
Sadly, I haven't been able to figure out what exactly the "degeneracies of entanglement witness eigenstates" are nor in what way they relate to our combinatorial context, much less how they are used in practice. So this is more of an exercise in name dropping than in explanation. I urge you to write me an email if you have explanations! On that same note, I feel it interesting enough to mention that the Catalan numbers can also be defined in an alternate and rather curious way. Instead of constructing them using factorials of integers (ie. binomial coefficients) the numbers can also be produced by calculating definite integrals over a particular trigonometric function. No need to trust my word though as you can try it out yourself on WolframAlpha. The function is somewhat reminiscent of the trigonometric definition of the proportionality constants that we looked into in the very first part of this series, but only somewhat.
In case you are not familiar with integrals there's an easy way to depict the calculation visually. We just need to plot the functions, which is usually done with x laid out horizontally from left to right and y growing from bottom to top. Once the function is drawn out we can state that the area left between the curve of the function and the horizontal x-axis over a chosen interval is equal to the value of an associated definite integral over that same interval. This definition is accurate enough for our purposes and actually performing the calculation is left as an exercise for enterprising readers. In our case we are only interested in the x-interval between zero and one (as indicated by the indices of our integral operator).
The diagram above shows the first four Catalan curves starting from n equals 0 and ending in n equals 3 — the same pair of numbers we used in our lattice example. The pleasant squiggly lines are meant to indicate the areas under investigation with all of the curves starting from their own relative origins where y equals zero. With the exception of the first curve all curves have two symmetric humps which means that they each contribute half of the integral value. All of the two-humped Bactrian curves are also all equal to zero at their midpoints because of contributing cosine. First curve makes an obvious exception due to its zeroth power. This completely orthogonal connection between a continuous (and awfully familiar) trigonometric integral and a bunch of discrete combinatorial integers is completely baffling to me and probably precisely for that reason also so marvelous. It is an open question – at least to me, at least for now – whether the Riordan numbers and the other combinatorial sequences we have identified have similar integral forms. I think this is a good note to end this particular adventure on.
What's next?
At this point of this series we have somehow managed to (loosely) tie together continued fractions, arbitrary optimal sections, the geometric progressions of proportionality constants, combinatorics and somehow even quantum physics. For hopefully obvious reasons this post will probably remain the high-point of this series. Still, there are some loose ends to take care of, so in the next part we will finally introduce a gentleman by the name of Chebyshev and see how his polynomials help us simplify the blueprints of optimal sections and their proportionality constants.
To be continued.