Happy Triangular Numbers

I have to confess to treating triangular numbers with some complacency over the years. Let's recall that triangular numbers are of the form:$$ \frac{n \, (n+1)}{2} \text{ with } n \geq 1$$The first triangular numbers are:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431

Figure 1: Triangular numbers appear in Pascal's Triangle.
In fact 3rd diagonal of Pascal's Triangle, gives all triangular numbers as shown

Similarly I've grown rather complacent about happy numbers that have the property that repeated iterations of the sum of digits squared lead to 1. For example:$$ 27261 \rightarrow 94, 97, 130, 10, 1$$The reason that I chose 27261 is that it's the number associated with my diurnal age today.

It turns out however, that numbers that are both triangular and happy are rather rare. These types of numbers comprise OEIS A076712 and the initial members of the sequence are:

1, 10, 28, 91, 190, 496, 820, 946, 1128, 1275, 2080, 2211, 2485, 3321, 4278, 8128, 8256, 8778, 9591, 9730, 11476, 12090, 12880, 13203, 13366, 13530, 15753, 16471, 17205, 17578, 20910, 21115, 21321, 22791, 24753, 25651, 27261, 29890, 30135, 31626, 33670, 35245

As can be seen, it will be quite some time before I meet the next such number: 29890. The main reason for creating this post was to draw attention to an interesting website titled Fascinating Triangular Numbers run by Shyam Sunder Gupta. It was begun on October 26th 2002 but remains active. There is a plethora of information on this site about various properties of triangular numbers so it's a wonderful resource.

What follows are just two examples from the site:

Example 1: The only known example of a Pythagorean triangle (\(a, b, c\)) where \(a, b, c\) are triangular numbers is (8778, 10296, 13530): $$ \begin{align} 8778^2 + 10296^2 &= 13530^2 \\ (\text{T}_{132})^2 + (\text{T}_{143})^2 &= (\text{T}_{164})^2 \end{align} $$Example 2: The only known examples of a Pythagorean triangle such that both Perimeter as well as Area are triangular numbers are: $$(3312, 14091, 14475) \\ \text{with Perimeter} = 31878 = T_{252} \\ \text{and Area}= 23334696 = T_{6831}\\. \\ \text{and} \\ .\\(3405996, 8013265, 8707079) \\ \text{with Perimeter}= 20126340 = T_{6344} \\ \text{and Area}= 13646574268470 = T_{5224284} $$