  This paper presents a study of the discrete Toda equation
$(\tau_n^t)^2+\tau_{n-1}^t\tau_{n+1}^t=\tau_n^{t-1}\tau_n^{t+1}$, that was
introduced in 1977. In this paper, it has been proved that the algebraic
solution of the discrete Toda equation, obtained via the Lax formalism, is
naturally related to the dual Grothendieck polynomial, which is a $K$-theoretic
generalization of the Schur polynomial. A tropical permanent solution to the
ultradiscrete Toda equation has also been derived. The proposed method gives a
tropical algebraic representation of static solitons. Lastly, a new cellular
automaton realization of the ultradiscrete Toda equation has been proposed.
