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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.

32
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8answers
5k views

Are Mersenne prime exponents always odd?

I have been researching Mersenne primes so I can write a program that finds them. A Mersenne prime looks like $2^n-1$. When calculating them, I have noticed that the $n$ value always appears to be odd....
19
votes
2answers
6k views

Closed form for $ S(m) = \sum_{n=1}^\infty \frac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$?

What is the (simple) closed form for $\large \displaystyle S(m) = \sum_{n=1}^\infty \dfrac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$? Notation: $ \dbinom{2n}n $ denotes the central binomial ...
13
votes
4answers
4k views

How to solve this sequence $165,195,255,285,345,x$

This is a question appeared in a competitive exam. The question is: Find the unknown term in $165,195,255,285,345,x$ 1)375 $\ \ \ \ \ \ \ \ $ 2)420 3)435 $\ \ \ \ \ \ \ $...
12
votes
4answers
43k views

Predict next number from a series

Which methods I can use to predict next number from a series of numbers ? I know the min & max possible number in advance.
6
votes
1answer
5k views

Find the pattern, What is the correct answer

Ok, so this question is from seriously hard IQ test that has been doing the rounds on facebook, ok, 17 questions are easy 3 are hellishly hard, The answer to this one is 17, I do not know why, I found ...
264
votes
13answers
10k views

Help with a prime number spiral which turns 90 degrees at each prime

I awoke with the following puzzle and I would like to investigate but the answer may require some programming (it may not either). I have asked on the meta site and believe the question to be ...
42
votes
17answers
31k views

Getting the sequence $\{1, 0, -1, 0, 1, 0, -1, 0, \ldots\}$ without trig?

What's the simplest way to write a function that outputs the sequence: {1, 0, -1, 0, 1, 0, -1, 0, ...} ... without using any trig functions? I was able to come ...
3
votes
0answers
117 views

Find palindromes in two consecutive number bases?

Can we generate all numbers which are palindromic in two consecutive number bases $(b, b+1)$ and have $(2d+1, d\in\mathbb N)$ digits when written in their palindromic bases? (I strongly ...
1
vote
1answer
62 views

Finding the $2n+1$ th derivative of $\frac{y^{2n+1}xy}{1-x^2y^2}$ with respect to $x$.

$f(x,y) = \frac{y^{2n+1}xy}{1-x^2y^2}$. I made the following table: \begin{align} & 2n+1 = 1 \implies f^{(1)} = \frac{1!y^2(1+x^2y^2)}{(1-x^2y^2)^2}\\ & 2n+1 = 3 \implies f^{(3)} = \frac{3!y^...
26
votes
2answers
478 views

What is this pattern found in the first occurrence of each $k \in \{0,1,2,3,4,5,6,7,8,9\}$ in the values of $f(n)=\sqrt{n}-\lfloor \sqrt{n} \rfloor$?

Learning how to generate the Mandelbrot set, I came across the definition of the "escape condition" which is the one that decides the color that is applied to each point of the plane where the ...
8
votes
2answers
338 views

Limit of the sequence $a_{n+1}=\frac{1}{2} (a_n+\sqrt{\frac{a_n^2+b_n^2}{2}})$ - can't recognize the pattern

Consider the sequence: $$a_0=x,~~~b_0=y$$ $$a_{n+1}=\frac{1}{2} \left(a_n+\sqrt{\frac{a_n^2+b_n^2}{2}} \right),~b_{n+1}=\frac{1}{2} \left(b_n+\sqrt{\frac{a_n^2+b_n^2}{2}}\right)$$ $$\lim_{n \to \...
6
votes
6answers
663 views

Do the differences of perfect squares apply to perfect cubes and more?

I'm curious about a special property of squares. The difference between perfect squares starting from 0 are 1,3,5,7,9..., where each difference goes up by 2. I want to know if there are any patterns ...
6
votes
1answer
334 views

Point Group of a pattern

I need to determine the point group of the following patterns: I think the one on the left is $D_1$ and the one on the right is $D_2$
6
votes
2answers
519 views

Formula to this pattern? $1$, $11$, $21$, $1211$, $111221$, $\ldots$

I have this pattern: 1 11 21 1211 111221 I'm guessing it's a fibo pattern, been at it for hours now. Anyone know?
3
votes
2answers
75 views

Is there an intuitive way to understand $Pair(x,y) = \frac{(x+y)(x+y+1)}{2} + x$?

After a lot of effort I discovered some pattern that the function: $$Pair(x,y) = \frac{(x+y)(x+y+1)}{2} + x$$ essentially zigzags along the grid with $\mathbf N$ vs $\mathbf N$ (natural numbers). So ...
2
votes
1answer
196 views

Is this a valid visualization of Euler's identity as a more generic pattern?

I was reading this nice question about a demonstration of Euler's identity, and tried to visualize how would look the left part of the identity in the complex plane by using the following function: ...
0
votes
3answers
7k views

What is the next picture in this sequence?

I was reading a book on logical reasoning and came across the following: Can anyone give me the answer to the question. We want the next sequence ?
-7
votes
5answers
2k views

How to solve the sequence: $87, 89, 95, 107, ?, 157$

This question appeared in a competitive exam. The question is: Q. Find the unknown term in $87,89,95,107,?,157$ 1)127 $\ \ \ \ \ \ \ \ $ 2)122 3)139 $\ \ \ \ \ \ \ \ $ ...
45
votes
1answer
1k views

Palindromic Numbers - Pattern “inside” Prime Numbers?

EDIT: rewritten and reduced entire post to present things more clearly. I'm asking how to calculate the next element(s) in the sequence, located at the end of this post. Introduction - ...
29
votes
1answer
484 views

A random walk on a finite square with prime numbers

This question is following two similar questions that you can find here and here. The idea is to walk on a square of length $n\times n$, following some rules. We will identify the opposite sides. ...
27
votes
3answers
593 views

A prime number random walk

This question came to my mind thanks to this question which I found really interesting (and beautiful! Like the mathematician Philippe Caldero said in his book Histoires Hédonistes de Groupes et de ...
11
votes
7answers
13k views

Pattern to last three digits of power of $3$?

I'm wondering if there is a pattern to the last three digits of a a power of $3$? I need to find out the last three digits of $3^{27}$, without a calculator. I've tried to find a pattern but can not ...
-5
votes
4answers
243 views

Summation of series 12,40,90,168,280,432,…? [closed]

I have been stuck in finding the general term and the sum of $n$ terms of the series $$12,\,40,\,90,\,168,\,280,\,432, ...$$ I am not able to see any relationship between the successive terms of the ...
6
votes
1answer
214 views

Please, help to identify this numerical constant

I'm trying to find an answer to this question. Let $K(k)$ be the elliptic integral of the first kind and $K'=K(\sqrt{1-k^2})$. According to Abel's theorem (see this link) we know that if $\frac{K'}{K}...
6
votes
2answers
655 views

Permutations to satisfy a challenging restriction

In a stack of n distinct cards in order {1,2,3,4,...,n} from top, define distance between 2 cards as the number of cards between them. 2 cards are neighbours if they're adjacent in original stack, ...
2
votes
0answers
78 views

Predict gaps between 9 digit 2-palindromes?

$p^2$ $\text{(or 2-palindrome) definition: }$ I call a palindrome $n$-palindromic or $p^n$ if it is palindromic in $n$ consecutive number bases. For example, $10$ is $p^2$ since it is palindromic in ...
1
vote
1answer
236 views

Patterns of no formula.

How do I find the next number if the given pattern is $$1,2,3,2,3,4,1,2,6,23,14,19,64,69,12,78,152,93,108,?$$ (Find the question mark)
-1
votes
3answers
2k views

Extract a Pattern of Iterated continued fractions from convergents

I have been working on an article at https://oeis.org/wiki/Table_of_convergents_constants where I posted a table of "convergents constants" (defined at https://oeis.org/wiki/Convergents_constant) for ...
6
votes
1answer
335 views

Crazy patterns arising from recursive sequence of functions

(It should first be stated that I'm all new to this kind of stuff, so do tell if anything turns out to be obvious to the more experienced reader, or even incorrect.) I've been considering the ...
3
votes
1answer
3k views

Bottom to top explanation of the Mahanalobis distance?

I'm studying Pattern recognition and statistics and almost every book I open on the subject I bump into the concept of Mahanalobis distance. The books give sort of intuitive explanations, but still ...
2
votes
2answers
4k views

How to use a difference table to find a formula for a given sequence?

Given a sequence, how do you get a "rule" from its difference table? For example; 1, 2, 3, 4, 5, 6 1, 1, 1, 1, 1 Or 1, 2, 7, 16, 29 1, 5, 9, 14 4, 4, 4 I haven't really explored this ...
1
vote
0answers
53 views

The $n$th derivative of the hyperbolic cotangent

While working on a problem, I came to this: What is the $n$th derivative of the hyperbolic cotangent? For simplicity, let $c=\coth(x)$. $c^{(0)}=c$ $c^{(1)}=-c^2+1$ $c^{(2)}=2c^3-2c$ $c^{(3)}=-...
1
vote
3answers
1k views

Collatz conjecture pattern (3n + 1 problem).

I have a pattern I found in Collatz Conjecture I want to share. Afterwards, I would like to know, if I could try harder at this pattern (I am stuck), if it could lead to a proof. Or it would just be ...
0
votes
1answer
4k views

Find the next number in this sequence

I have a homework. It seems to be an easy sequence but I can't get the answer. So, What is the next element? $$2, 7, 10, 13, 23, 34,?$$ What would be the solution out of these numbers? 45 or 49 or 58 ...
-4
votes
1answer
139 views

$n$ such that the digits immediately after the decimal point of $\pi^n$ give $n$ again

I was doing something with value of $\pi$ as I know that the beauty of numbers will always exist , doesn't matter either number is real or complex it must be beautiful. I observe something strange by ...
5
votes
0answers
388 views

Unexpected pattern in consecutive 7 digit “double” palindromes?

Why does this branching structure appear in consecutive 7 digit "double" palindromes? Dots at the end of the rows indicate that I excluded following values for that row (because of the character ...
3
votes
3answers
2k views

Largest number on multiplying with itself gives the same number as last digits of the product

What is the largest number on multiplying with itself gives the same number as last digits of the product? i.e., $(376 \times 376) = 141376$ i.e., $(25\times 25) = 625$ If the largest number cant ...
3
votes
2answers
156 views

What is the $2012th$ number in this pattern?

This is question 30 from Australian Maths 2012 $(0,1,2,1,2,3,2,3,4,1,2,3,2,3,4,3,4,5,2,3,4,...)$ What is the $2012th $ number in this list? What I did: I broke up the first few numbers into ...
2
votes
1answer
751 views

Pattern finding for repeating sequences

Find a pattern for the following sequence. 1,1,3,1,3,5,7,1,3,5,7..... Let n= the nth number. Find the nth number of this series
2
votes
3answers
130 views

Is there a way of making “guess the next number in the sequence” rigorous?

This is maybe more of a question for matheducators.SE than math.SE but I'm more interested in the math than the education. A common problem given to middle and high school kids (at least in America) ...
1
vote
2answers
345 views

What is the relation between this binary number with no two 1 side by side and fibonacci sequence?

I saw this pattern of binary numbers with constraints first number should be 1 , and two 1's cannot be side by side. Now as an example ...
0
votes
2answers
44 views

See the pattern in the following product involving e

I've the following pattern, how do I write it in product form:  1. $$e-\text{k}\cdot\text{x}_{\text{k}-1}$$  2. $$e-\text{k}\cdot\left(e-\left(\text{k}-1\right)\cdot\text{x}_{\text{k}-2}\right)$$  3. ...
0
votes
1answer
1k views

Pattern matching questions

I saw the below questions and was curious regarding the logic behind the answer. I will be grateful is some one could help me So the question is given the patter, identify the next one
0
votes
2answers
99 views

Find the missing Item in the sequence [closed]

There is a sequence described below.What is the missing item? $5,10,?,50,122,...$
0
votes
0answers
46 views

Help with formula for complicated pattern

This picture represents 56 Lunar years. On the Left side, you have the months, not twelve but 13 ( you can go here for an explanation on that) - The numbers highlighted in orange above all the ...