I rediscovered (or maybe discovered) another obscure way to find the fibonacci sequence lol

I seem to do that a lot

Do tell.

yes

I am interested

Unfortunately, the margins of chat are too small for Redwolf to say :(

The pattern of zeroes in these fractions

I'd explain where they're from but it takes a lot of context and is mostly pointless

It involves the collatz conjecture

@RedwolfPrograms oh no, go oon

I have all day

literally, I have the next 4 1/2 hours to hear this

not in OEIS!

Simplified

@RedwolfPrograms @RedwolfPrograms elaboration is required

Basically, imagine there are two functions, we'll call them f and g. f(x) = x / 2, and g(x) = 3x + 1.

ok

Now, take all of the different permutations of f and g where there are no adjacent gs, and the first and last function is f

Examples: f, fgf, ffgf, ...

Now, solve an equation f(x) = x, or f(g(f(x))) = x, or f(f(g(f(x)))) = x

These are the results

lol, wtf

well, now I know where the numbers come from

With the permutations of f and g being represented by integers in binary, where f is 1 and g is 0

but why the fibonacci sequence

I know why but it's complicated to explain

What was the motivation behind this?

So these factions are the solutions? How are they ordered?

@RedwolfPrograms again, I have all day

@WheatWizard With the binary representations of the permutations of f and g, with the first fraction there being f (1)

oh

So the zeros are just where the function is just a chain of fs right?

Yep

And whenever you have sequences without adjacent 0s in their binary representations, the fibonacci sequence tends to pop up

Yeah. Neat

Interestingly, if you could prove there was never a fraction there that was a positive integer (with a few additional properties), it would prove that there were no numbers that led to infinite loops (rather than continuing to infinity or stopping at 1) for the collatz conjecture

well, you should submit that to the folks trying to prove th ecollats conjecture

I doubt it's useful

How are you generating the fractions? Just brute forcing?

I'd submit it to the OEIS but I have absolutely no idea how to explain what this is

@WheatWizard I found a really cool way of doing it

I wrote an arbitrary precision rational library really quickly, so that's Rational in this code to generate it:

var n_ND = (number) => {
    var n = 0n;

    for (var i = 1n; n < number; i++)
        if (i % 2n && !i.toString(2).includes("00"))
            n++;

    return i - 1n;
};

var r_ND = (number) => {
    var ND = n_ND(number).toString(2);

    var a = Rational();
    var b = Rational();

    for (var i = 0; i < ND.length; i++)
        [a, b] = ND[i] == "0" ? [a.add(1).multiply(3).add(-1), b.multiply(3).add(1)] : [a.add(-1).divide(2), b.divide(2)];

    return b.multiply(-1).divide(a);
};

Thanks markdown

I'd explain that but there's ghosts in it

Ok yeah that's what I meant by brute forcing. In retrospect there are even more brute force ways to do it.

Basically if you keep track of an equation ax+b=0, you can just do a really simple tranformation of a and b for each f or g, then the result is just -b / a

n_ND (sorry for the awful naming) just finds the permutation of f and g at that index

sagecell.sagemath.org/?q=iqiapx

I seem to be getting different results

You're not filtering the indices

That's what n_ND does in mine

You can't have two gs next to each other

Or end with a g

oh

The end with f restriction is better to not handle with a filter.

I don't actually filter the indices, I just find the nth valid index

The valid indices are A247648

hm? It looks like you are filtering in your code. But idk what language this even is.

JS

ಠ_ಠ I love it when requiring case insensitivity (rather than just guaranteeing single case input) doubles my byte count

I'll write a nicer version of this that's actually meant for other people to read

Well the point is: if you count up by twos you don't need to check that it ends in a 1, since odd numbers are numbers that end in 1.

Oh, that's what you meant. Good point.

Nicer version, with Rational included

@RedwolfPrograms that's because you can append a 1 at any time i.e. in F(n-1) ways but you can only append a 0 if it does not already end in 0 i.e. you appended a 1 last time, which is F(n-1-1) ways, thus F(n) = F(n-1) + F(n-1-1) where F(n) is the number of such sequences

(Even better you can just take a normal binary number , chop off the head and alternate it with 1s e.g. if you want the n is 100010 then the nth number is 10101011101)

@RedwolfPrograms the zeros are at 0, 1, 3, 6, 11, 19, how do you get fibonacci out of that?

probably the gaps between them

If you shift around indexes and stuff you get it

@hyper-neutrino ah

@WheatWizard This is basically O(1), but you probably don't care too much about the performance.

@rak1507 I get 1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, 609, 986, ..., which appears to be fibonacci(x) - 1

sagecell.sagemath.org/…

fun

I don't really know why I spent two hours on this lol

i mean the consecutive differences between the fibonacci numbers are the fibonacci numbers like... by design. lol

Yeah, that's sort of like "all primes with less than two factors" :p

@RedwolfPrograms well, this is interesting

the prefix sum array of fibonacci is just fib - 1 as well, unsurprisingly. :P

fibonacci is recursive

well you can define it with a closed form formula too

@hyper-neutrino which is?

A closed (form)²ula?

@StackMeter The Binet formula

i C now

Wait, I wonder what that looks like on the complex plane.

Annoyingly desmos does not do that

Maybe I can hack it together with a parametric graph

Oh wait if you google complex fibonacci numbers and go to google images you can see it lol

Keep it. That is not worth fixing.

I got 69 errors again :\

@RedwolfPrograms My code never is

69 errors is too perfect

what lang, scala?

Yeah

I got 69 errors yesterday too

hexagony

Hexagony's taken

it's the only good one

6-gon

It's going to use base 6 instead of base 2 for the integers

Maybe Heximal?

hexagondwana land

hexagone with the wind

I like that one :P

"Frankly, my dear, I don't give a pointer"

If you like verbose names, you could name it A language related to a certain polygon (hint: the number of sides the polygon in question has is approximately equal to twice the ratio between the circumference of a circle and its diameter)

narthexagon

@user 12?

hexmix

Yes

haruspexagon

complexagon

@RedwolfPrograms You didn't see anything :P

I've had the idea for this language for like a year now

The interpreter's just going to be annoying

Good name

I support this name too

but it's probably already taken

I want to implement a language which is encoded on a system with a non-regular base, e.g base 2 in some places and base 3 in some other places.

Ahh I'm at 78 errors now

:(

@WheatWizard Would they be randomly chosen?

@user When it comes to errors big numbers are closer to 0 than small numbers. Fixing the problem usually involves creating more errors first.

@user Looks like most of those were just the formatter being dumb and removing my imports, luckily

@user I don't think so. Probably some non-repeating predictable pattern.

I'm thinking maybe Thue-Morse, since it is self similar.

@WheatWizard this is already in most languages. (decimal for source, binary for runtime)

I would really hate decimal if I didn't use it so much

That's not what I mean at all.

:)

why base 3

why not base 27

you at least have to beat decimal

CMQ: Favorite base?

Mine's definitely heximal

hmmm

32

because there are 32 symbols on a keyboard

however, second fav is 27

Base 6 is 100% the best base IMO

Easily represents halves, thirds, quarters, fifths, sixths, sevenths, eights, ninths, and so on

you mean 60

No, I mean 6

0.1111... is 1 / 5

6 is only good for 2, 3, and 6

Yeah, but it's almost good for 5 and 7

@RedwolfPrograms -1+i

@RedwolfPrograms Does water count? :P

fair

@RedwolfPrograms cats. Also, 10

For decimal you only get 2, 5, and 10 free, and 3, 9, and 11 are almost good

my fave base is however many symbols there are in unicode that don't relate to a letter

@Wezl This?

Some people like base 12. I dislike those people.

the Babylonians used base 60

Oh, you like base 12? :P

@StackMeter Wait what

I vote for base 720 or 3600

Twelve is awful at representing fifths, which automatically invalidates it

I like base 10 (glares at people who don't get it)

No one told me the Babylonians had 60 fingers

@StackMeter Good luck remembering that many symbols

@user no seriously

@Wezl Without a subscript, base 10 is implied

Smaller bases are more efficient

Six is also good for counting by hand

@RedwolfPrograms lol read it and cry

Yeah, like base 1. When everything is 0, math is easy

Zero to five fingers = zero to five in heximal

Base 6 is the best base

No debate

@RedwolfPrograms if you have to bring two hands out, you might as well use all ten fingers

@user also then sed can handle it

@StackMeter You don't!

6 digits, 6 fingers, ceil(6/5) = 2 hands

You can count from 0 to 5, and you can naturally count up to 35 with two hands

@Wezl By base 1, I mean a positional system, not tally marks

@StackMeter You're doing it wrong then

[0, 1, 2, 3, 4, 5]

.-.

@RedwolfPrograms how so

@RedwolfPrograms You can count up to way more than that if you use your hands properly

Steal someone else's hands and start counting on those

Base 6 wouldn't have a special symbol for 6, so why should you be able to hold up six fingers for 6?

Just hold up zero on one hand and one on the next

@RedwolfPrograms damn I fooled myself

@RedwolfPrograms Isn't it called hexal?

You can count from 0 to 35 that way

@WheatWizard I've heard like 20 names for it

Senary, seximal, heximal, hexal, etc.

@RedwolfPrograms Oh yeah, that makes sense

I think the m in decimal comes from decem.

I guess to be consistent it should be sexal.

And binary should be dual

> A senary (/ˈsiːnəri, ˈsɛnəri/) numeral system (also known as base-6, heximal, or seximal)

Why not make them all m-ary instead?

Decimal could be deciary or something like that

I like heximal because senary sounds weird and it doesn't make immature people like me laugh

Unlike seximal

hexary, decary

nonary

@WheatWizard What's kaiary?

:clueless:

I mixed up my ancient greek.

Decary looks weird, I feel the "c" should be soft

kai is a conjunction.

Deciary might be nicer

Oh ok

I don't usually read ca with a soft c.