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m = 121; (* number of terms required *)

S[1] = {1};

S[n_] := S[n] = SubstitutionSystem[{1 -> {1, 2}, 2 -> {3}, 3 -> {1, 2}}, S[n-1]];

For[n = 2, True, n++, If[PadRight[S[n], m] == PadRight[S[n-1], m], Print["n = ", n]; Break[]]];

Take[S[n], m] (* Jean-François Alcover, Feb 15 2023 *)

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This sequence is the [0->12, 1->3]-transform of the Fibonacci word A003849: if T(0):=12, T(1):=3, then one proves easily with induction that T(phi_1^n(0)) = phi-hat_2^{n+1)(}(1), and T(phi_1^n(1)) = phi-hat_2^{n+1)(}(2), where phi_1 denotes the Fibonacci morphism given by 0 --> 01, 1 --> 0. - Michel Dekking, Dec 29 2019

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Kevin Ryde: (Mismatched braces vs parens.)

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