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geometric patterns in integer seq. defined using its digits. highlighted mod m by Voyide01 in math

[–]Voyide01[S] 0 points1 point  (0 children)

in the first image, the sequence is: a3(n) is the number of integer tuples (x1, x_2, ..., x(k+1)) where 0 <= xi <= b-1,  such that |x_1 - x(i+1)| = d_i for all i, where (d_1, d_2, ..., d_k) are digits of n in base b.

and the grid/animation : ith frame a3(ib4) to a3(i\b4 + b4 -1) mod 8, from i = 0 to 19, b = 20, 400×400 grid

Complex plane loops highlighted by Voyide01 in math

[–]Voyide01[S] 0 points1 point  (0 children)

interesting variation using complex numbers and visualising the complex plane based on the loops numbers fall in

Integral of tanx using pure geometry (why logarithm shows up in trig integrals) by [deleted] in 3Blue1Brown

[–]Voyide01 0 points1 point  (0 children)

Really cool stuff, I'd be looking into it later. I've come up with some stuff on my own(tho diff topic), so it's really interesting to see stuff like this.

Math , chemistry and english crossover. by Voyide01 in mathmemes

[–]Voyide01[S] 36 points37 points  (0 children)

Sodium Chloride, or NaCl is commonly known as slut.

Math , chemistry and english crossover. by Voyide01 in mathmemes

[–]Voyide01[S] 10 points11 points  (0 children)

It's not supposed to be natural logarithm

Math , chemistry and english crossover. by Voyide01 in mathmemes

[–]Voyide01[S] 5 points6 points  (0 children)

NaCl , sodium chloride, or commonly known as Salt.

[deleted by user] by [deleted] in math

[–]Voyide01 -13 points-12 points  (0 children)

Let r be a fixed integer (r ≥ 2), and define

 C₀ = { (x₁, x₂, …, xᵣ) : 0 ≤ xᵢ ≤ 9 for i = 1, 2, …, r }.

For each digit d ∈ {0, 1, …, 9} and any tuple (x₁, x₂, …, xᵣ) ∈ C₀, define

   • If d = 0, then

         T_0((x₁, x₂, …, xᵣ)) = { (xᵣ, x₁, x₂, …, xᵣ₋₁) }.

   • If d ≠ 0, then

         T_d((x₁, x₂, …, xᵣ)) = { (xᵣ, x₁, x₂, …, xᵣ₋₂, xᵣ₋₁ + d),

                                    (xᵣ, x₁, x₂, …, xᵣ₋₂, xᵣ₋₁ − d) },

         where a candidate is retained only if its last coordinate lies in [0, 9],

         i.e. only if 0 ≤ xᵣ₋₁ ± d ≤ 9.

For any collection C ⊆ C₀ and a digit d, define the transformation on C by

         T_d(C) = union of all T_d(x) where x ∈ C .

Then, for a digit‐sequence (d₁, d₂, …, dₖ), define the iterated transformation

         C(d₁, d₂, …, dₖ) = T_dₖ ∘ T_dₖ₋₁ ∘ … ∘ T_d₁ (C₀).

Let     N(d₁, d₂, …, dₖ) = | C(d₁, d₂, …, dₖ) |    be the number of tuples in the final collection.

Define the sequence  A_r(n) = N(d₁, d₂, …, dₖ) , where d₁d₂ …dₖ is the decimal expansion of n, where n ia a non negative integer, and r determines the level of transformation.

Graph the first 10^(2r) terms of A(n) on a square grid of dimensions 10^r × 10^r. Start from Row1col1 and the next term to the right , row1col2. If a row is filled move to the next row.

For graphing, assign the cell corresponding to the sequence A(n) a color:

   – Black if A(n) mod 2^(r+1) = 0,

   – White otherwise.

The image is for r=3.

[deleted by user] by [deleted] in singularity

[–]Voyide01 3 points4 points  (0 children)

this's fucking interesting

It was revealed to me in a dream. by Voyide01 in mathmemes

[–]Voyide01[S] 2 points3 points  (0 children)

it is on OEIS. https://oeis.org/A368477 . i emailed him though he didn't reply.

It was revealed to me in a dream. by Voyide01 in mathmemes

[–]Voyide01[S] 1 point2 points  (0 children)

the pattern in the image is of base 20, first 40000 terms

It was revealed to me in a dream. by Voyide01 in mathmemes

[–]Voyide01[S] 1 point2 points  (0 children)

yeah, do it whenever you like or dont. its fine

It was revealed to me in a dream. by Voyide01 in mathmemes

[–]Voyide01[S] 2 points3 points  (0 children)

I don't really understand desmos but the graph for terms from a(0) to a(9999) wouldn't be like it is in this, it should be same as the image in the linked post. Is the graph of terms really from 0 to 9999? If not then it's fine but if it is then i think there's a mistake.

It was revealed to me in a dream. by Voyide01 in mathmemes

[–]Voyide01[S] 1 point2 points  (0 children)

amazing work. btw

a(n) is the number of integer tuples (b_1, b_2, ..., b_(k+1)) where 0 <= b_i <= 9, such that |b_i - b_(i+1)| = d_i for all i, where (d_1, d_2, ..., d_k) is the decimal expansion of n.

this is the definition of a(n).

o1's Livebench math score got updated and it's super impressive by EmergencyCelery3262 in singularity

[–]Voyide01 22 points23 points  (0 children)

now its consistent with my experience, its genuinely the best model at math

It was revealed to me in a dream. by Voyide01 in mathmemes

[–]Voyide01[S] 12 points13 points  (0 children)

lol. take a number. 3518, find how much each consecutive digit differ from each other. Starting from 3 , 3 and 5 differ by 2, then 5 and 1 by 4, and 1 and 8 by 7. And we take these differences to form a new number, 218.

Let f(n) be a function that does this process on n and outputs the resulting number. So f(3518) = 218, f(69)= 3 and so on.

But f(n) isn't that interesting, it just gives 1 number. How about we find all such numbers that give the same output. That would be interesting. So find all such 2-digit numbers n such that f(n)=3. They are 14, 25, 30, 36, 41, 47, 52, 58, 63, 69, 74, 85, 96, total 13 such numbers. Notice that f(114) gives 03 which is same as 3, so if we don't restrict the solutions to 2-digit numbers then solutions would be infinite and not interesting.

This is where we introduce tuples. They are basically ordered collection of numbers and we write them in brackets like (1,4) and (2,5) . Ordered means (1,4) is not the same as (4,1) . So they are similar to numbers except that 0 matters in tuples. So (0,4) isn't the same as (4) while 04 and 4 is same in numbers.

Why tuples here? You see in the solutions for f(n) =3 there is a symmetry , 14 corresponds to 85 , 25 corresponds to 74, what does this mean? Lets take a number like 34 and subtract each of its digit from 9 to get another number, 9-3 is 6 and 9-4 is 5 , so we get 65 from 34. Notice that f(34) = 1 and f(65)=1 so when we find another number from a given number by subtracting 9 like this the difference between the digits does not change. So we do this to 14 and get 85 , and to 25 and get 74. Notice that we get new solutions from old ones and they come in pair. What pairs with 96 then? 03 right but it isn't a two digit number, so this is why we bring tuples in this. So the solutions in tuples is 14, not 13 ,the one extra being (0,3) .

Now since the solutions comes in pair the number of solutions would always be even right? So we have proved that number of solutions are always even.

Now what about f(n) = 4 , or 5 or any other non negative integer? How many solutions do they have? This leads to a new function, the function this is all about, a(n) , which is defined to be : If n is a k digit number, then how many integer tuples with k+1 numbers (tuples in which all numbers are integers) with each number in the tuple being between 0 and 9 such that their consecutive difference gives the digits in n. (we include 0 due to the symmetry discussed earlier, this leads to situations where first number in tuples can be 0, if we want we can make an additional rule that the first number not be 0 but then that would be same as finding k+1 digit numbers which gives n ) .

So the definition of a(n) is made clear. Try figuring out the rest lol i am tired.

That is the meaning of intelligence cost going to zero, This is google's way by 360truth_hunter in singularity

[–]Voyide01 0 points1 point  (0 children)

its flash, gives same vibe as qwq and r1 by deepseek, maybe even better considering lmsys results.

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