When are Squares Triangles?

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This is based upon the final part of this discussion (2) for children given by Rav Ginzburgh on the relationship of square and triangle numbers in the structure of Torah and his paper, "When Two Triangles Make a Square.

{∆, ∎, o} = {
{8, 6, 36},
{49, 35, 1225},
(288, 204, 41616),
...}

We define the Triangle operator, ∆, and Square operator, ∎, as follows:

∆ ≣ { n(n+1)/2 = m^2 = o = ∎ <=> m = o^(1/2) = (n(n+1)/2)^(1/2) | m, n, and o ∈ N }

Then:
∆ = (n(n+1))/2 = m^2
∎ = m^2 = o
=> ∆ =  ∎ = o

Using the original values we will uncover the underlying function that sends ∎_i to ∆_k and ∆_i to ∎_k.
{∆_i, ∎_i, o_i} = { 
{8, 6, 36},
{49, 35, 1225},
{288, 204, 41616},
...} 

(∎_1 - 1) * (∎_1 + 1) = o_1 - 1 = ∎_1^2 - 1

*_*_*_*_******** ^ ********_*_*_*_*

This is the correct formula to find the rest of the values:

(o_2 - 1) / ∎_1 = ∎_3
(o_3 - 1) / ∎_2 = ∎_4 = 1189
(o_4 - 1) / ∎_3 = ∎_5 = 6930
(o_5 - 1) / ∎_4 = ∎_6 = 40391
(o_6 - 1) / ∎_5 = ∎_7 = 235416
to infinity...

∆_1 = 8
∆_2 = 49
∆_3 = 288
∆_4 = 1681
∆_5 = 9800
∆_6 = 57121
∆_7 = 332928

∎_1 = 6
∎_2 = 35
∎_3 = 204
∎_4 = 1189
∎_5 = 6930
∎_6 = 40391
∎_7 = 235416

o_1 = 36
o_2 = 1225
o_3 = 41616
o_4 = 1413721
o_5 = 48024900
o_6 = 1631432881
o_7 = 55420693056
etc... 

*_*_*_*_******** v ********_*_*_*_*

∴ (o_j - 1) / ∎_i = ∎_k
for all of the values derived from the table and holds ∀ i, j, k ∈ N; 1 ≤  i < j < k < ∞. []
--------

From comments in the video:
∎_1 * (∎_2 - 1) = ∎_3

Note: This exception apparently produces only one sequentially accurate result:
∎_2 * (∎_3 - 1) ≠ ∎_4 => 7105 ≠ 1189

however since 7105^2 = 50481025 gives n = 10047 which does resemble the pattern, although it is not in this sequence:
=> ∎_2 * (∎_3 - 1) = ∎_x

∎_3 * (∎_4 - 1) = 242353, 
242353^2 = 58734976609 => n = 342740
etc...
--------

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