The integer 2-domination formula for three-row grids

The conjecture holds for EVERY positive integer $n$, with an exact finite-state induction certificate.

Fix the height $h$, let $\mathcal C=\{0,1,2\}^{h}$, put $\mathbf0=(0,\ldots,0)$, and write $|b|=\sum_i b_i$. For $a,b,c\in\mathcal C$, extend $b$ by $b_0=b_{h+1}=0$ and define

$$Q(a,b,c)\iff a_i+b_{i-1}+b_i+b_{i+1}+c_i\ge2\quad(1\le i\le h).$$

This is exactly the closed-neighborhood constraint for the middle column. Let $V_n(a,b)$ be the minimum weight of an $n$-column labeling with last columns $a,b$, all columns before $b$ already valid, and the fictitious left boundary zero. Then

$$V_1(a,b)=\begin{cases}|b|&a=\mathbf0,\\+\infty&a\ne\mathbf0,\end{cases} \qquad V_{n+1}(b,c)=|c|+\min_{a:Q(a,b,c)}V_n(a,b).$$

The exact domination number is obtained by adjoining the zero right boundary:

$$\gamma_{\{2\}}(G_{h,n}) =\min_{a,b:Q(a,b,\mathbf0)}V_n(a,b).$$

The time-independent transfer operator $T$ satisfies $T(V+d)=T(V)+d$. Therefore an identity of COMPLETE state vectors $V_{N}=V_{N-p}+d$, including every state, implies by induction

$$V_{n+p}=V_n+d,\qquad \gamma_{\{2\}}(G_{h,n+p})=\gamma_{\{2\}}(G_{h,n})+d \quad(n\ge N-p).$$

For $h=3$, direct exact evaluation gives the COMPLETE 729-coordinate identity

$$V_{19}(a,b)=V_{15}(a,b)+6 \qquad(a,b\in\{0,1,2\}^{3}).$$

For widths $1,\ldots,18$, the terminal optima are respectively

$$2,4,5,7,8,10,11,13,14,16,17,19,20,22,23,25,26,28.$$

They equal $\lfloor3n/2\rfloor+1$. The state-vector identity propagates the recurrence $\gamma_{\{2\}}(G_{3,n+4})=\gamma_{\{2\}}(G_{3,n})+6$ to ALL $n\ge15$, and the asserted formula obeys the same recurrence. Therefore

$$\boxed{\gamma_{\{2\}}(G_{3,n})=\left\lfloor\frac{3n}{2}\right\rfloor+1\quad\text{for every }n\ge1.}$$

For completeness, the entire exact finite certificate is reproducible with the following standard-library-only code. Missing dictionary entries represent $+\infty$; the final cardinality checks verify that BOTH certificate vectors contain ALL 729 states.

from itertools import product

h, N, period, increment = 3, 19, 4, 6
C = tuple(product(range(3), repeat=h))
Z = (0,) * h
weight = {c: sum(c) for c in C}

def valid(a, b, c):
    return all(a[i] + b[i] + c[i]
               + (b[i-1] if i else 0)
               + (b[i+1] if i+1 < h else 0) >= 2
               for i in range(h))

successors = {(a,b): tuple(c for c in C if valid(a,b,c))
              for a in C for b in C}
V = {(Z,b): weight[b] for b in C}
history = {}
for n in range(1, N+1):
    history[n] = V
    optimum = min(v for (a,b),v in V.items() if valid(a,b,Z))
    assert optimum == 3*n//2 + 1
    W = {}
    for (a,b),v in V.items():
        for c in successors[a,b]:
            state = (b,c)
            W[state] = min(W.get(state, 10**9), v+weight[c])
    V = W
A, B = history[N-period], history[N]
assert len(A) == len(B) == len(C)**2
assert A.keys() == B.keys()
assert all(B[state] == A[state]+increment for state in A)
print("PASS:", h, len(B), N, N-period, increment)

The full-vector identity and min-plus homogeneity supply the induction for arbitrarily large $n$; the finite computation is a complete induction certificate, not extrapolation from sampled grid sizes.