Questions tagged [pattern-recognition]

1. From a samples of a small samples of mathematical objects, conjecture a common pattern to all of them. This includes "guess the next terms in the sequence" question (consider checking OEIS first). Please provide as much context as possible. 2. Mathematical ideas related to pattern recognition, subfields of AI and statistics. Please check first if StackOverflow, Computer Science Stack Exchange, or Cross Validated is more appropriate.

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Strange pattern in numbers why is the series of absolute differences always 0? [duplicate]

Put any four numbers in a row, say, 77 73 29 179. Under the first number write the difference between it and the second number. Under the second number put the difference ...
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When are fractions of factorials equal to whole factorials? [duplicate]

10!/7!=6!. 6!/3!=5!. Is there something interesting happening in these examples, or are they just coincidence? Is there a way of finding other examples like these?
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Fix my irregular spiral

I recently posted a puzzle featuring a graphic of a stylized rosette. The rosette is nothing more than a rectangular spiral. See here for the original puzzle post. Here is a distillation of the ...
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Hidden patterns in the natural numbers

Consider the matrix $\{a_{ij}\}_{i,j\in\mathbb{N}}$ with $a_{ij} = 1$ iff $i \equiv 0\,\text{mod}\,j$. Drawing it with stretched y-axis by a factor of $5$ reveals some "waves" which I ...
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Pattern in number of digits in $(10^n)!$

I was observing the OEIS sequence A061010 and its values up to $n=1000$, and it seems that the pattern is as follows. For $n$, $f(n)$ is for the form string of $(n-1)$+(pattern of repeating digits ...
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A "mysterious" sequence?

A recent question asked for the Taylor expansion of $$f(x)=\exp\left(-\pi\,\frac {_2F_1(\frac12, \frac12, 1; 1-x)}{_2F_1(\frac12, \frac12;1;x)}\right)=\exp\left(-\pi\,\frac{K(1-x)}{K(x)}\right)=\frac{...
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How to design a clustering metric or cluster algorithm to seperate tow intersected circles?

In some classical books about pattern recognition, the clustering algorithm reveals the similarity among samples, while not considering the "global pattern" of grouped samples. So, I think ...
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1answer
25 views

How to find the general formula to calculate the slope of a line inside the polygon?

I want to find the general formula to calculate the slope of the bold line in the picture when we put one side of the regular $n$-gon upside. I have found $3$ data below, even though I haven't ...
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Recursive problem - Staircase

I I am working on recursive formulas and I managed to solve the fib formula and got it to F(0) = 0 F(1) = 1 F(N) = F(N - 1) + F(N - 2) Now I stumbled on another ...
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$1-8-28-21-4-32-17-19-30-6-3-13-12-24-25-11-5-31-18-7-29-20-16-9-27-22-14-2-23-26-10-15-1$ - amazing circle [duplicate]

I found an amazing circle while surfing on the internet.(I will call this circle a 32-AC.) It is a circle with 32 numbers with no numbers written twice. Also, the sum of any two adjacent numbers is a ...
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partly piecewise increasing Floor function, ask for value in certain condition

A Maths Olympiad problem we have $f(n) = \lfloor\frac{n}{\lfloor\sqrt{n}\rfloor}\rfloor$ try find for all n (and prove) such that $f(n)< f(n+ 1)$, $f(n) > f(n+ 1)$, $f(n)= f(n+ 1)$. with n ...
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Where is my logic flawed (IQ Test question)

The iq test in question - image 1 Available answers - image 2 I solved this IQ test questions and later found my answer was wrong. I selected F when the correct answer is B. I do understand how B can ...
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Scale invariance prior distribution. Prove the following statement.

Consider a density of the form $$ p(x|\sigma)=\frac{1}{\sigma}f(\frac{x}{\sigma}). $$ where $\sigma>0$. Note that this will be a normalized density provided $f(x)$ is correctly normalized. ...
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What is intuition behind writing the explicit formula for a sequence?

I am working through some sequence problems and am a bit stuck. We are told to write the explicit formula for two sequences and then find $a_{100}$ and $a_{20}$ respectively. The two sequences are the ...
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Climbing an infinite staircase whilst becoming predictably faster and slower

During a study of game theory, I discovered this problem which I'm not sure how to solve: On an infinite staircase, a person is able to travel 1 step/second faster every 500 stairs until they climb ...
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Find the $n$'th integer composition with a fixed number of components in lexicographical order

lets there is a number (N) and partitions (p), I have all the possible combinations, each combination sums to N. I want to calculate the nth term of these sequences/ combinations given N and p. N = 7, ...
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997 views

Find an equation of the sequence $1, 3, 12, 60, 360, 2520, \cdots $. [closed]

Proof: By rule of formation $$a_1=1=\frac{(1+1)!}{2}$$ $$a_2=3=\frac{(1+2)!}{2}$$ $$a_3=12=\frac{(1+3)!}{2}$$ $$a_4=60=\frac{(1+4)!}{2}$$ $$a_5=360=\frac{(1+5)!}{2}$$ $$a_6=2520=\frac{(1+6)!}{2}$$ $$\...
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Optimal sequence horizon for predicting the next iteration

Suppose a mechanism generates numbers from 1 to 6, say, one at the time. What is the best way to examine if a particular sequence of numbers has a higher probability of leading to ‘1’, say, as the ...
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Why does quality factor $q_i$ in Relevant Vector Machine represent a measure of the alignment of $\boldsymbol{\phi_i}$ with the error of the model?

In RVM(relevant vector machine, also named Sparse Bayesian Learning), there are two important factors $s_i$ and $q_i$, the $q_i$ has been explained as 'quality factor'. In Bishop's PRML section 7.2.2 ...
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Patterns in indefinite integral (1)

So far, I've found out this: $\int \arcsin x \mathrm d x = x \arcsin x + \sqrt{1-x^2}+C$ $\int x\arcsin x \mathrm d x = \dfrac{x^2}{2}\arcsin x +\dfrac {x}{4}\sqrt{1-x^2} -\dfrac{1}{4}\arcsin x + C$ $\...
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Unproved patterns

Let $z_1,z_2,z_3\in\mathbb{C}$ such that $|z_1|=|z_2|=|z_3|=1, z_1+z_2+z_3=1$. Calculate $z_1^{2003}+z_2^{2003}+z_3^{2003}.$ Let's say $S_k=z_1^k+z_2^k+z_3^k$. I tried calculating $S_2,S_3,S_4,S_6$ so ...
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Derivative with Chain Rule of Sum of Squares Error Function when fitting the data using Mth order polynomial

This is from the very first exercise of Chris Bishop's Pattern Recognition and Machine Learning. The questions asks us to show how minimizing the Sum-of-Squares Error Function through the weights w is ...
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Why do we use $...$ with irrational numbers?

That $...$ in title means the $...$ used at the end of example in question body. So my question is you must have numbers/patterns like $$ W={1,2,3 ...} $$ Here $ .. $ makes sense. Also $$ \frac{1}{3}=...
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Finding total unique combinations and Generalisation of the same.

There are n numbers of k bits, I want to find out how many unique combinations will be if a bit position is set for each of the ...
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What is the seeming pattern here?

So I can see that to go to the input from step 1 involves a addition of 7, which is the 4th term. But then I can't figure out any link to the 2nd or other steps. Any help would be much appreciated. ...
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1answer
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Remainder of number divided by ${4^n}$ must be half divisor

I am a student in programming and observed a pattern in a sequence. I wondered if it is a well-known principle so I can optimize the algorithm. I am not a mathematician, so I hope this will make sense....
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1answer
33 views

A solution to this two-dimensional recurrence equation

I have a table of values that follows a specific pattern. The right and bottom edge are all ones. Let the table be indexed from bottom up and rigth to left, i.e 0,0 is at bottom right. A value at ...
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General formula for a sequence defined by a recursive system

Let $M=(\alpha_{k,n})_{(k,n)\in \mathbb{N}}=\begin{pmatrix} \alpha_{0,0}& & & (0) \\ \alpha_{1,0}&\alpha_{1,1} & & \\ \alpha_{2,0}&\alpha_{2,1} &\alpha_{2,2} & \\ \...
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1answer
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How can I approximate a sine function using neural networks?

I am trying to replicate the figure 5.9 of the book Machine Learning and Patter recognition. But in the figure the output function is a soft one, in my architecture it is just joined points. How can I ...
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What is the closed form for the sequence $1,0,1,1,0,0,1,1,1,0,0,0...$? [closed]

I thought about $$\frac{1+(-1)^k}{2}$$ but it didn't work ! Is there any closed form for it ?
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Probability density over a thin shell of radius r of multivariate gaussian

I am following Pattern Recognition and Machine Learning by Bishop and in problem 1.20 we try to find the probability density of a thin shell of radius $r$ of a multidimensional gaussian. The ...
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Relating product of univariate gaussian integrals to surface area of unit sphere

In Pattern Recognition and Machine Learning by Bishop exercise 1.18, the author states the following relation: $\prod_{i=1} \int_{-\infty}^{\infty} e^{-x_{i}^2} dx_{i}= S_{D} \int_{0}^{\infty} e^{-r^2}...
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1answer
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Any pattern in the following sequence [closed]

I am practicing finding patterns and sequences with numbers and found one that I cannot seem to find the logic. $$ \frac{2}{3} ,\frac{4}{3} ,\frac{4}{5} ,\frac{6}{5} ,\frac{6}{7},\frac{6}{8}... $$ The ...
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Find the series $\sum A_{k}$ if $A_{k} = |a_{2^{k}N}| + |a_{2^{k}N+1}| + \cdot \cdot \cdot + |a_{2^{k+1}N-1}| $

Pick a natural number $N \geq 1$. For $k \geq 0$, where $A_{k}$ is defined as $$A_{k} = \left( |a_{2^{k}N}| + |a_{2^{k}N+1}| \right) + \left( |a_{2^{k}N+2}| + |a_{2^{k}N+3}| \right) + \cdot \cdot \...
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Greatest number of occurrences of the pattern 4213 in a permutation.

Statistic St000750 in the FindStat database is a map $\operatorname{St000750} \colon S_n \rightarrow \mathbb N_{\geq 0}$ given by The number of occurrences of the pattern $4213$ in a permutation. ...
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Finding general term of sequence

I need to find the general terms of the sequence, $$\{1.25, 2, 3.25, 5, 7.25, \cdots\}$$ and $$\{0, 6, 24, 60, 120, 210, \cdots\}$$ I can't think of any such pattern that would produce these.
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Showing that losing positions in Wythoff's game are generated by $(\lfloor n\phi\rfloor, \lfloor n\phi^2\rfloor)$, where $\phi$ is the golden ratio

A very neat problem in combinatorial game theory: In Wythoff's Game, how do I show that all the losing positions are generated by the formula $$\left(\lfloor n\phi\rfloor, \lfloor n\phi^2\rfloor\...
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1answer
33 views

Find the pattern of a series

I am given the series $$\frac{1}{6} + \frac{1\cdot 8}{6\cdot 10} + \frac{1\cdot 8\cdot 15}{6\cdot 10\cdot 14}$$ I am asked to find a formula for this series $a_n$ So I have found that the last ...
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Coordinates reachable from $(x, y) = (2,5)$ [closed]

If we are on coordinate $(m,n)$ then we can either go to coordinate $(m + n, n)$ or $(m, m + n)$. If we start at $(2,5)$ then what all coordinates we can reach in finitely many moves? In our case we ...
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28 views

Find the Pattern/Sequence

(1+2) (1+2)(4+2) (1+2)(4+2)(7+2) (1+2)(4+2)(7+2)(10+2) I can see that each term in the sequence is multiplied by an additional term that is 3 greater than the previous 2 terms. How can I represent ...
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1answer
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In Fisher’s discriminant for multiple classes, How do you manage when $(Sw)$ is singular matrix (so you cant get $(Sw)^{-1}$)?

I am trying to use Fisher’s discriminant for multiple classes to reduce the Dimension of the MNIST data set, similar to this post: https://towardsdatascience.com/an-illustrative-introduction-to-...
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1answer
30 views

Identifying specific pattern in a noisy dataset

I am working on a project where I have the following forward model. There are multiple point sources sparsely distributed in 3D space with coordinate $[x_0,y_0,z_0]$ for each of them. The detected ...
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1answer
264 views

Which is the algorithm for knn density estimator?

I am reading Pattern Recognition and Machine Learning by Christopher Bishop. In chapter two he talk about using knn to density estimation. I want to replicate a plot using python/R/matlab. He is ...
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105 views

Finding nth term for a recursive/iterative/term to term sequence

I have the sequence: 1, 16, 166, 1666 . Im trying to find the nth term for this sequence but since it is not linear or geometric I'm not sure how to. I worked out that the recursive formula is: \begin{...
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The numbers in these patterns go together following the same rule. Find the missing number.

I don't see the pattern of the numbers, can someone tell me what the pattern is?
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1answer
49 views

Finding a general expression for a recursively defined sequence.

I have a sequence of numbers that I would like to find a general expression for. The sequence is defined as follows. $$C_m=\frac{1}{2m-3}\left[C_{m-1}+\frac{m^2-9m+16}{9\cdot 2^{m+4}(m-4)!}\right], m\...
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Is there any value related to a time series pattern?

I've been digging a little into signals and I just started a networks and telecommunications class at college. We saw there are different encode techniques Amplitude, Frequency and Phase shift keying (...
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16 views

Function to classify 3 classes using two class classifier

I am reading machine learning on my own. I am thinking of single function to classify 3 classes using 2 class classifier. Here is $f(x) \geq 0, g(x) \geq 0 \implies C_3$ $f(x) \leq 0, h(x) \geq 0 \...
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2answers
38 views

Need help figuring out a pattern question

my cousin asked for help on this math problem and the answer is apparently 6 but i can't find a coherent pattern that fits all of the triangles. I included an image of the question.
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1answer
56 views

Higher Order Euler Differential Equations

What is the pattern for writing the indicial equation for higher order Euler Ordinary Differential Equations? For first order Euler ODEs, the indicial equation coefficients mirror the ODE coefficients ...

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