Articles of pattern recognition

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 where I posted a table of “convergents constants” (defined at 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} […]

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 see one? Am I missing something? Thanks for your help in advance!

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 Géométries (roughly translated) “Let us stop for a moment to contemplate the beauty of mathematics, that is after all the point of figures”.). […]

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. Formally, the square with the opposite sides identified can be modelled by $(\mathbb Z/n\mathbb Z)^2$. We will walk following the […]

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 $\ \ \ \ \ \ \ \ $ 4)140 I tried very much to solve it but can’t find the correct answer. Any hints will […]

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 ?