Completely Random Sequence

Discrete Mathematics Level pending

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.

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