Problem : Generational Triplets (Not commenting on tutorial blog since community discourages necroposting)

Before seeing the editorial, I observed the following:

1. Let a satisfying sequence be represented by a pair $$$(a, d)$$$ [first term and common difference respectively], with $$$\gcd(a, d) = 1$$$. Call such a sequence a basis.
2. For a given $$$d$$$, only one such $$$a$$$ can exist. (Proof)
3. Using the basis we can form all "related" sequences as follows: $$$(a, d)$$$ $$$\rightarrow$$$ $$$(2a, 2d)$$$ for all $$$a$$$ and $$$(a, d)$$$ $$$\rightarrow$$$ $$$(2a + 1, 2d + 1)$$$ when $$$a$$$ is odd.

Is it possible to get to a complete solution using these observations (provide counter-example or justification if you think any of these is wrong although most of them I have proven)?