# Pascal’s Triangle

2016-03-26, post № 111

mathematics, programming, Pygame, Python, #crown, #generate, #sequence

Pascal’s triangle is an interesting mathematical sequence. It is often written as a triangle, starting with $\{1\}$, then $\{1,1\}$. To generate the next row, you add the two numbers above to form another one. So the next row in the sequence is $\{1,2,1\}$ then $\{1,3,3,1\}$, $\{1,4,6,4,1\}$ and so on (sequence A007318 in OEIS).

One interesting property of Pascal’s triangle is the generation of binomials.
To calculate $(a+b)^4$, you can look at the 𝟦-th row (listed above and when starting to count at 𝟢) and determine

This program generates Pascal’s sequence in a rather unusual shape, looking a bit like a crown.

## Controls

• ‘Space’ takes a screenshot.
Source code: pascals-triangle.py

[1]

# Palindromic Primes

2016-03-23, post № 110

mathematics, programming, Python, #generating, #numbers, #palindrome, #palindromes, #palindromic numbers, #prime numbers, #prime palindromes, #symmetric numbers, #symmetry

TheOnlinePhotographer has published a post to celebrate 𝟣𝟩𝟣𝟩𝟣𝟩 comments and was amused by the number’s symmetry.
A great comment by Lynn pointed out that this number is indeed an interesting number but not symmetrical.
Symmetrical numbers or words — also called palindromes — are defined as being the same read forwards or backwards. Examples for palindromic words are “radar”, “noon” or “level”. Palindromic numbers are 𝟥, 𝟦𝟢𝟦 or 𝟣𝟩𝟤𝟤𝟩𝟣.

Lynn then went further and checked if 𝟣𝟩𝟣𝟩𝟣𝟩 is at least a prime [1]. The number sadly has five distinct prime factors ($171717=3\cdot 7\cdot 13\cdot 17\cdot 37$).

So Lynn wondered what the next palindromic prime would be.
To answer this question, I wrote this little Python program to check for palindromic primes. The first 𝟣𝟤𝟢 palindromic primes are shown below.
Based on this list, the smallest palindromic prime larger than 𝟣𝟩𝟣𝟩𝟣𝟩 is 𝟣𝟢𝟢𝟥𝟢𝟢𝟣.

      3,       5,       7,      11,     101,     131,     151,     181,
191,     313,     353,     373,     383,     727,     757,     787,
797,     919,     929,   10301,   10501,   10601,   11311,   11411,
12421,   12721,   12821,   13331,   13831,   13931,   14341,   14741,
15451,   15551,   16061,   16361,   16561,   16661,   17471,   17971,
18181,   18481,   19391,   19891,   19991,   30103,   30203,   30403,
30703,   30803,   31013,   31513,   32323,   32423,   33533,   34543,
34843,   35053,   35153,   35353,   35753,   36263,   36563,   37273,
37573,   38083,   38183,   38783,   39293,   70207,   70507,   70607,
71317,   71917,   72227,   72727,   73037,   73237,   73637,   74047,
74747,   75557,   76367,   76667,   77377,   77477,   77977,   78487,
78787,   78887,   79397,   79697,   79997,   90709,   91019,   93139,
93239,   93739,   94049,   94349,   94649,   94849,   94949,   95959,
96269,   96469,   96769,   97379,   97579,   97879,   98389,   98689,
1003001, 1008001, 1022201, 1028201, 1035301, 1043401, 1055501, 1062601, ...

Thus it takes $1003001-171717=831284$ more comments to reach the closest palindromic prime.

The sequence of palindromic primes is number A002385 in the On-line Encyclopedia of Integer Sequences (OEIS).

Source code: palindromic-primes.py

# RGB Color Cube

2016-03-19, post № 109

programming, Pygame, Python, #16777216, #256**3, #all rgb, #blue, #colors, #every rgb color, #green, #red

Listing the red part on the 𝑥-axis, the green part on the 𝑦-axis and the blue part on the 𝑧-axis, this program displays all the 𝟣𝟨𝟩𝟩𝟩𝟤𝟣𝟨 [1] colors rgb can display.
The 𝑧-axis is shown over time, visually going through a cube of colors.

Source code: rgb-color-cube.py
Jonathan Frech's blog; built 2021/04/16 21:21:49 CEST