Completely Random Sequence

Are there any infinite sequences of numbers that are not defined by a finite number of symbols from the English alphabet?

For example: $1, 104, 4372, 96256, 1240002, ...$ is a sequence whose definition is the expansion of $\frac{16(1+k^2)^4}{(k\cdot k'^2)^2}$ in powers of $q$ where $k$ is the Jacobian elliptic modulus, $k'$ the complementary modulus and $q$ is the nome.