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Pascal's triangle has wide applications in Mathematics.I have seen that the most important applications relate to the binomial coefficients and combinatorics.

Are there some other beautiful interesting patterns in Pascal's triangle that can be found by selecting some other combination modes ?

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closed as primarily opinion-based by Najib Idrissi, Daniel Fischer Aug 31 '16 at 10:34

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Search in Google Images writing "Pascal's Triangle". $\endgroup$ – user 170039 Aug 30 '16 at 13:51
  • $\begingroup$ Also see this. $\endgroup$ – user 170039 Aug 30 '16 at 13:52
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    $\begingroup$ Why does the title ask about applications but the question ask about patterns? There are many applications that make use of the binomial coefficient properties one way or another, like e.g. describing a deterministic Galton board. Conversely, current answers tend to investigate patterns without neccessarily naming applications for them. $\endgroup$ – MvG Aug 30 '16 at 22:33
  • $\begingroup$ I suppose if the questions asked for actual applications, as the title suggests, then it would've been more well-received... $\endgroup$ – Wojowu Aug 31 '16 at 11:03
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How about Fibonacci sequence in the Pascal Triangle? enter image description here

Pascal's Triangle (from Wolfram MathWorld)

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There is a nice hexagonal property in the Pascal triangle:

\begin{array}{ccccccccccccccccc} &&&&&&&&1\\ &&&&&&&1&&1\\ &&&&&&1&&2&&1\\ &&&&&1&&3&&3&&1\\ &&&&1&&4&&6&&4&&1\\ &&&1&&\mathbf{\color{blue}{5}}&&\mathbf{\color{blue}{10}}&&10&&5&&1\\ &&1&&\mathbf{\color{blue}{6}}&&\mathbf{\color{red}{15}}&&\mathbf{\color{blue}{20}}&&15&&6&&1\\ &1&&7&&\mathbf{\color{blue}{21}}&&\mathbf{\color{blue}{35}}&&35&&21&&7&&1\\ 1&&8&&28&&56&&70&&56&&21&&8&&1\\ \end{array} The hexagon around $\binom{n}{k}$ fulfils \begin{align*} \binom{n-1}{k-1}\binom{n}{k+1}\binom{n+1}{k}&=\binom{n-1}{k}\binom{n}{k-1}\binom{n+1}{k+1}\\ \end{align*}

The example $\binom{n}{k}=\binom{6}{2}=\color{red}{15}$ shows \begin{align*} \binom{n-1}{k-1}\binom{n}{k+1}\binom{n+1}{k}&=\binom{5}{1}\binom{6}{3}\binom{7}{2}=\color{blue}{5}\cdot \color{blue}{20}\cdot \color{blue}{21}=2100\\ \binom{n-1}{k}\binom{n}{k-1}\binom{n+1}{k+1}&=\binom{5}{2}\binom{6}{1}\binom{7}{3}=\color{blue}{10}\cdot \color{blue}{6}\cdot \color{blue}{35}=2100 \end{align*}

In other words:

  • A hexagon around a binomial coefficient $\binom{n}{k}$ is always a perfect square. :-)

See Generalized hidden hexagon squares by A.K. Gupta for a generalized version of this relationship.

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    $\begingroup$ That's a truely remarkable property: where else could you find hexagons that are perfect squares? $\endgroup$ – leftaroundabout Aug 30 '16 at 21:18
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Pascal’s triangle reduced modulo $2$ is a discrete version of the Sierpiński gasket; this page is a very brief introduction, and this page at OEIS has a nice picture and some more information.

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    $\begingroup$ This JS app I made will allow you to visualize the triangle under various different moduli. Enter modulo = 2 to get the version mentioned in this answer. For higher moduli, different colors represent different congruence classes, and monochrome mode shows zero congruence classes in black, and everything else in white (or possibly the opposite, I can't remember). The study of these diagrams easily reduces to the prime modulus case, for which there are some interesting patterns. $\endgroup$ – Jack M Aug 30 '16 at 22:09
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The one I like the most is associated to a very well known primality test: The number $n \gt 1$ in the second position (left or right) of a given line of the triangle is a prime number if and only if all the elements of the same line, except the first and last $1$'s, are divisible by it.

For instance:

$1,5,10,10,1$

$n=5$ and $5 \mid 10$, so $5$ is a prime number.

In other hand for instance:

$1,4,6,4,1$

$n=4$ and $4 \not\mid 6$ so $4$ is not prime.

$1,6,15,20,15,6,1$

$n=6$ and $6 \not\mid 15$, $6 \not\mid 20$ so $6$ is not prime as well.

All the elements must be divisible by $n$, if just one fails, it is not a prime number.

So performing the $\pmod{n}$ operation, the pattern at right side shows the rows of a prime $n$ as follows:

enter image description here

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    $\begingroup$ Nice observation! (+1) $\endgroup$ – Markus Scheuer Aug 31 '16 at 8:55
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    $\begingroup$ So could you say the following: a number $n\ge2$ is prime iff $\sum_{i=0}^n (\binom{n}{i}\pmod n) = 2$ or iff $n \mid (\sum_{i=0}^n\binom{n}{i})-2$ $\endgroup$ – QBrute Aug 31 '16 at 10:23
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    $\begingroup$ Fun fact: this fact is a basis for how AKS primality testing algorithm (the first deterministic, provably polynomial-time general purpose primality testing algorithm) works. $\endgroup$ – Wojowu Aug 31 '16 at 10:28
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Here's a way to extend Pascal's triangle; I think it's interesting.

In black we have the original Pascal's triangle. In green we have inserted rows between the black rows. To find the values, we use the polynomials which describe the slanted columns of the original triangle, shown in blue, and evaluate at half integers. Pascal plus

So what's the point of this? Remember the numbers in Pascal's triangle also provide the binomial coefficients.

For example, $(1+x)^4=1+4x+6x^2+4x^3+x^4$.

These new rows also describe the coefficients of the series for binomials raised to the half integer power: $(1+x)^\frac{7}{2}=1+\frac{7}{2}x+\frac{35}{8}x^2+\frac{35}{16}x^3+...$

Unlike the rows of Pascal's triangle, these continue forever. Also, like Pascal's tringle, the green numbers are the sum of the two green numbers directly above. For example, $\frac{15}{8}+\frac{5}{16}=\frac{35}{16}$.

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  • $\begingroup$ Nice answer! (+1) $\endgroup$ – Markus Scheuer Aug 31 '16 at 8:55
  • $\begingroup$ I feel like the term "binomial series" should at the very least be mentioned in the answer. $\endgroup$ – Wojowu Aug 31 '16 at 10:08
  • $\begingroup$ My intuition tells me that this works for any complex x - producing the series expansion of the power operation. EDIT: @Wojowu that's exactly what I meant with that :) $\endgroup$ – WorldSEnder Aug 31 '16 at 10:18
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Here are some interesting facts about Pascal's triangle compiled from Google search.

enter image description here

enter image description here

enter image description here

enter image description here

enter image description here

enter image description here


All hail Google...

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    $\begingroup$ The first image (gif) looks like babble to me. Looks like its author is trying to build some new theory out of it... Does anyone know whether it really makes sense? $\endgroup$ – Wojowu Aug 31 '16 at 10:13
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    $\begingroup$ The same seems true for the second image; people building "new mathematics" or "new physics" often amuse me. $\endgroup$ – Wojowu Aug 31 '16 at 10:24
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    $\begingroup$ @Wojowu Blame the author. $\endgroup$ – Anastasiya-Romanova 秀 Aug 31 '16 at 10:50

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