We realize the $\mathrm{SU}(3)$ Verlinde sums $V_n(m)$, up to anexplicit factor, as the values at $n$ of the spectral zeta functionof a higher-order Laplace operator on thetriangular discrete torus on $m^2$ vertices. For fixed $m$, we express their generating function in terms of thelogarithmic derivative of an associated even spectral polynomialand express this polynomial as an explicit iterated resultant.Exploiting permutation symmetry, we prove that this polynomial isa cube over $\mathbb{Q}$, apart from an explicit quadratic factorwhen $3\mid m$.This factorization yields shorter linear recurrences satisfied by $V_n(m)$ with constantcoefficients.We also derive Binet-type formulas expressing $V_n(m)$ as finitelinear combinations of powers of rescaled inverse squares of theroots of the spectral polynomial.These results provide an efficient algorithm for computingthese Verlinde sums.Several fully developed examples demonstrating computational efficiency of the method are given, including expressions in terms of Fibonacci and Lucas numbers.