Solution: We are to count the number of sequences of length 5 (one per layer), where each element is an epoch from 1 to 4, and no two consecutive layers have the same epoch. - Parker Core Knowledge

Understanding the Context

In this article, we explore a classic combinatorics problem: Counting sequences of length 5 where each element is an integer from 1 to 4, and no two consecutive elements are equal. The solution applies dynamic counting principles useful in algorithm design and system configuration.

Problem Statement

Count the number of valid sequences of length 5, where each element in the sequence is an integer from 1 to 4 (inclusive), and no two consecutive elements are the same. This models, for example, epoch choices across 5 training stages with restricted repetition.

Key Insights

Formally, we want the number of sequences:
(a₁, a₂, a₃, a₄, a₅)
such that:

• aᵢ ∈ {1, 2, 3, 4} for all i = 1, 2, 3, 4, 5
  
• aᵢ ≠ aᵢ₊₁ for all i = 1, 2, 3, 4

Approach: Recursive Dynamic Counting

Let’s denote Aₙ(k) as the number of valid sequences of length n where the last element is k, and k ∈ {1, 2, 3, 4}. Since all values from 1 to 4 are symmetric in constraints, Aₙ(k) will be the same for each k.

Step 1: Base Case

For n = 1 (first layer), any of the 4 epochs is allowed:

• A₁(k) = 1 for k = 1, 2, 3, 4
  So total sequences: T₁ = 4 × 1 = 4

Final Thoughts

Step 2: Recurrence Relation

For n > 1, when building a sequence of length n ending with k, the previous layer (n−1) must be any value except k. Since there are 4 possible values and one is excluded (k), there are 3 valid predecessors.

Thus:
Aₙ(k) = sum_{j ≠ k} Aₙ₋₁(j) = 3 × Aₙ₋₁(1)
But since all Aₙ₋₁(j) are equal, say x, then:
Aₙ(k) = 3x
And total sequences:
Tₙ = sum_{k=1 to 4} Aₙ(k) = 4 × 3 × Aₙ₋₁(1) = 12 × Aₙ₋₁(1)
But Aₙ₋₁(1) = Tₙ₋₁ / 4 (since all end values are equally distributed)

Substitute:
Tₙ = 12 × (Tₙ₋₁ / 4) = 3 × Tₙ₋₁

Thus, we derive a recurrence:
Tₙ = 3 × Tₙ₋₁, with T₁ = 4

Step-by-Step Calculation