What if we had discovered triangular numbers before multiplication was invented? I came to this question after I found an identity that connects triangular numbers

\[ \Delta(x) = \frac{x(x-1)}{2} \]

with multiplication, namely

\[ x y = \Delta(x + y) - \Delta(x) - \Delta(y). \]

To see why this is true, just draw the large triangle with edge length of \( x + y \) and remove the two smaller triangle from it — then a lozenge with \( x y \) points remains.

So if we had a pocket calculator with a \( \Delta \) key instead of a multiplication key, then we could still do all integer calculations that involve only addition and multiplication. But what about algebra? Here my results are mixed so far. When trying to translate the commutative law

\[ (x y) z = x (y z) \]

into triangular arithmetic, I got a long formula which provided no insight. But associativity,

\[ x (y + z) = x y + x z \]

is equivalent to the nice formula

\[\begin{align}
&\Delta(x + y + z) - \Delta(x + y) \\
&{} - \Delta(y + z) - \Delta(x + z) \\
&{} + \Delta(x) + \Delta(y) + \Delta(z) = 0.
\end{align}\]

#Mathematics #Arithmetic #NumberTheory #TriangularNumbers

Consecutive Squares as Up/Down Sum of Odds

#Algebra #Math #MathematicalVisualProofs #Mathematics #nowWatching #NumberTheory #oddNumbers #proof_ #squares #triangularNumbers #YouTube

#OEIS #TriangularNumbers

[Continued 1]

A triangle of numbers (keyword 'tabl' in the OEIS) isn't particularly sexy, nor are the generating functions usually used to describe them formally.

What is unfortunately done far too rarely is to represent the triangle using an interpolating analytical function, although this is not so infrequently possible.

In the case of triangular numbers, it is easily possible, and it starts with seeing that the representation has several symmetries, which become apparent when extended to the entire ZZ x ZZ plane.

.., 9, 6, 4, 3, 3, 4, 6, 9, ..
.., 7, 4, 2, 1, 1, 2, 4, 7, ..
.., 6, 3, 1, 0, 0, 1, 3, 6, ..
.., 6, 3, 1, 0, 0, 1, 3, 6, ..
.., 7, 4, 2, 1, 1, 2, 4, 7, ..
.., 9, 6, 4, 3, 3, 4, 6, 9, ..

Looking at the function in a graphical representation, for example on Wolfram Alpha, you can see that it has a pretty visualization: It resembles an ellipsoid vessel.

If you divide this vessel into eight equally sized parts, you can mark the triangle on it. The classic triangular numbers are located on four ascending lines.

#OEIS #TriangularNumbers

Triangular numbers. The ancient Greeks were among the first to study them systematically, but they are also found in Egyptian and Chinese mathematics.

Triangular numbers are numbers that can be represented as an equilateral triangle. Each number T(n) in the sequence is the sum of all positive integers up to n. For example, the first few triangular numbers are 0, 1, 3, 6, 10, etc.

The question we ask is: How can the triangular numbers be generalized? Our answer will again generate a triangle, which seems only appropriate, but this time from triangle numbers.

To do this, we write the triangular numbers one below the other as a column in the first step. At each level, we branch off a path to the right, the same length as the path we just traveled, but start counting again at the branching point (i.e., from 0).

Carl Friedrich supposedly found the formula for triangular numbers in elementary school, but the recursive formulation is more interesting:

Starting with T(0) = 0, we define T(n) = n + T(n-1).

For our extension, this becomes the recurrence:

T(n, n) = n + T(n, n - 1) starting with T(0, 0) = 0.
For k != n: T(n, k) = n + T(n-1, k).

We must still convince ourselves that the result also meets our intuitive expectations.

(1) If you look at the resulting triangle, you can see that the columns and the diagonals bring together the who's who of the figurative numbers. We have hit a sweet spot.

(2) But what does the formula look like? It couldn't be simpler:

T(n, k) = T(n) + T(k).

Can we confirm that these numbers have been studied for millennia (references, please) and have been in the OEIS for decades?

Well, at least the latter is not the case. https://oeis.org/A367964

Luke: What's a 1-by-1 square, what's a step squad up to 2, what's a 3-by-3 square, what's a step squad up to 4, what's a 5x5 square, and what's a step squad up to 6?
Me: 🤔 🧠 🤨 1, 3, 9, 10, 25, 21
Luke: 21? The step squad up to 6 is not bigger than the 5x5 square?
Me: Nope.

He wanted to see it for himself so I got out the graph paper.

Luke: This one's 21. And this one's 25. But this seems bigger.

It took some re-counting before he become convinced!

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