Articles of pattern recognition

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

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) is something like “Find the next number in the sequence $2,4,8,16,\dots$”. Now I am not against this problem, finding patterns […]

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 be found out can you prove that there is always a number greater than any given number? (only in base […]

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 or 39

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 $c=\coth(x)$. $c^{(0)}=c$ $c^{(1)}=-c^2+1$ $c^{(2)}=2c^3-2c$ $c^{(3)}=-6c^4+5c^2-2$ $c^{(4)}=24c^5-34c^3+10c$ Etc. It appears to be representable as a polynomial of $c$. Any ideas on what the coefficients are? Update: It appears the leading coefficient is trivially given by […]

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 1 = 1 10 = 1 100,101 = 2 1000,1001,1010 = 3 10000,10001, 10010, 10100, 10101 = 5 Strangely I see the numbers we can form of this […]

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 a few numbers. It would be nice to support the article with some quality analysis. Before June 9, 2011, was starting to extract and clearly define a pattern to these constants cf the article. […]

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, if their index differs by 1. How many card rearrangements (permutations) satisfy the following? I) The sum of distances between any […]

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

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 series. Is there a pattern which I am not able to see?

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: $$a_0=x,~~~b_0=y$$ $$a_{n+1}=\frac{1}{2} \left(a_n+\sqrt{\frac{a_n^2+b_n^2}{2}} \right),~b_{n+1}=\frac{1}{2} \left(b_n+\sqrt{\frac{a_n^2+b_n^2}{2}}\right)$$ $$\lim_{n \to \infty} a_n=\lim_{n \to \infty} b_n=l(x,y)$$ I can’t pin down the pattern for the limit. Numerically, I’ve got the following result: $$\frac{1}{l(x,y)}=\arctan (f(x,y))$$ What is the explicit expression for $f(x,y)$? Numerical examples: $$\begin{array}( x & y & \frac{1}{l(x,y)} \\ 1 & 2 & \arctan \left( \frac{3}{4} […]