Basic math
1+3 =4   ,         4 is 2 squared
1+3+5=9  ,       9 is 3 squared
1+3+5+7=16,  16 is 4 squared
and it continues on forever...



The Fibonacci sequence is another well-known pattern where each number is the sum of the two preceding ones, usually starting with 0 and 1:
0,1,1,2,3,5,8,13,21,…


Arithmetic sequences have a common difference between terms, while geometric sequences have a common ratio.

    Arithmetic: a,a+d,a+2d,a+3d,…a,a+d,a+2d,a+3d,…
    Geometric: a,ar,ar2,ar3,…a,ar,ar2,ar3,…


Prime numbers have the property that they are divisible only by 1 and themselves. There are various patterns and conjectures related to primes, such as the Twin Prime Conjecture, which posits that there are infinitely many "twin primes" (pairs of primes that differ by 2, like 11 and 13).


Pascal's Triangle is a triangular array of binomial coefficients. Each number is the sum of the two numbers directly above it. It has numerous applications in probability, combinatorics, and algebra.


Fractals like the Mandelbrot set and the Sierpinski triangle exhibit self-similar patterns at every scale.


eix=cos⁡(x)+isin⁡(x)eix=cos(x)+isin(x) beautifully connects complex exponentials with trigonometric functions, and its special case eiπ+1=0eiπ+1=0 is considered one of the most elegant equations in mathematics.

These are just the tip of the iceberg. The world of mathematics is rich with patterns, some of which are still not fully understood!