Sequence-based Math System

[Viska]

Greetings, fellow number lovers! This is my first post to this group, it's nice to meet you all!

I'm a big fan of OEIS as it has been vital to my R&D. I have developed a math system that generates and assists in identifying sequences by operating out of base x which turns equations into numbers. I just recorded a 3-hour introduction to my system and wanted to share it with an appreciative audience.

https://x.com/ViskArbo/status/1985408008860389850?t=-FSuw9Z7n7BQRW9IbcLynA

One key result of this is identifying sequences through an abc/xyz format. For example a sequence can be defined as: 

add 3 if even index, double if odd index, start from s_0 = 7, then the sequence is 

{7,10,20,23,46,49,...} And then by performing analysis we can identify this as 

7.10,-1,-7 / 1.0,-3,0,2. In base 10 this becomes 

7.983 / 0.9702 but it's hard to read the decimal output because the sequence grows so quickly so use the aerated version 

7.0,10,0,-1,0,-1 / 1.0,0,0,-3,0,0,0,2 in base 10 is 

7.099893 / 0.99970002 - put that in your calculator it's 

7.10202346...

The fact that the denominator was already aerated indicates that the even and odd index sequences are distinct and can be seen as unrelated. We can de-weave them extracting odd vs even indexed numbers in numerator: 

7.10,-1,-7 / 1.0,-3,0,2 = 

7.0,-1 / 1.0,-3,0,-2 + 

0.10,0,-7 / 1.0,-3,0,2 

then pop it all as it's all aerated, then we get 

7.-1 / 1.-3,2 for even index (7,20,46,...) and 

10.-7 / 1.-3,2 for odd index (10,23,49,...).

1.-1 × 1.-2 = 1.-3,2 (9×8 = 100-30+2 = 72)

7.-1 ÷ 1.-1 = (69 / 9 = 7.66666)

7.6,6,6,6,6,6 ÷ 1.-2 = (69/72 = 

7.20,46,...

10.-7 ÷ 1.-1 = (93/9 = 10.333333)

10.3,3,3,3,3,3,3 ÷ 1.-2 =  (93/72 = 

10.23,49,...

The numerator can be viewed as the input and denominator / multiplier as a function acting on the input. Watch the video to learn more, would love to hear feedback! I have much much more to share. Thanks!

[Charles Greathouse]

If I’m understanding correctly, you’ve discovered rational generating functions (equivalently, linear recurrence relations with constant coefficients). They’re pretty useful! Every sequence in the OEIS which has a line like

<a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).

can be understood in these terms. They’re pretty useful have nice properties, for example, interleaving two such sequences results in a third with the same property.

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[Fred Lunnon]

  Good catch, Charles --- I failed to spot this. 

  An internet search for "rational approximation" is one way into this topic, which 

apparently goes back to Bernoulli in 1730, although not systematically developed 

until 1890 by Henri Padé --- see 

  https://mathshistory.st-andrews.ac.uk/Biographies/Pade 

The questioner joins what I suspect is a long roll-call of researchers to rediscover 

it in ignorance, which incidentally includes myself.  

  The "Padé table" is a kind of difference table on steroids, its entries "rational 

functions" (quotients of polynomials in one variable), and constructed via a simple 

recurrence credited to J. J. Sylvester; this breaks down when it encounters multiple 

zero denominators ("Regge poles" in physics applications, so I understand). 

  In the 1970's, Marcel Froissart and I independently extended the recurrence to cope 

with coincident poles, via a truly improbable identity which (for all I know) remains 

inscribed for perpetuity on the toilet wall of John Conway's former home; and surely 

merits a blue plaque? 

  See also the Mathologer video "Secrets of the Lost Number-walls" 

    https://www.youtube.com/watch?v=NO1_-qptr6c 

(btw, it lasts a good deal less than 3 hours ... ). 

WFL 

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[Viska]

Excellent! Thanks so much for pointing this out. I've had a hard time finding other materials akin to what I've been doing, so I'm excited to dig into this :)

I find this system to be prophetically simple, effective, and revealing, especially when contrasted with algebraic f(x) definitions, as the algebraic / letter components can be defined by position instead of explicitly (100 = 1 vs x² = 1) which reduces many operations to translation and super-positioning, and just about everything can be represented and acted upon as number sequences which cuts overhead significantly.

Thanks again!

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[Allan Wechsler]

Viska, probably at some point you should check out Herbert S. Wilf's generatingfunctionology, a comprehensive textbook on how to look at sequences as the coefficients of power series. The book has been made available for free online by the author, at https://www2.math.upenn.edu/~wilf/gfology2.pdf .

As far as your notation, representing polynomials as if they were numbers "in base x", the main problem is that these polynomials look too much like ordinary numbers. My instinct is to put a subscript "x" after such abbreviations, in analogy with putting the radix in a subscript. This is hard to write in plain text, but perhaps you could use Mathematica's underscore convention ("100_x") or the square-bracket convention from many programming languages ("100[x]"). The notation has its limitations: large, non-integral, or negative coefficients are hard to express. But I can certainly imagine cases where the notation would pay for itself.

-- Allan

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