I heed help regarding a Discrete math problem?

Let state 0 represent “start, or just wrote 0 or 2”. Let state 1 represent “just wrote 1”. Then the 2-by-2 transition matrix T representing allowable transitions is (in Mathematica notation) T = {{2,1}, {2,0}} and the number of allowable sequences of length n is given by the matrix product {1,0}.

T?. {1,1} {1,0}. {{2,1}, {2,0}}?.

{1,1}. Now T can be diagonalized as T = EDE? ¹ where E = {{½(1+?3), ½(1-?3)}, {1, 1}} D = {{1+?3, 0}, {0, 1-?3}} E?

¹ = {{1/?3, ( 3 -?3)/6}, {-1/?3, ( 3 +?3)/6}}. So the number of sequences of length n can be computed as {1,0}. ED?

E? ¹. {1,1} 1/6((3 - 2?3)(1-?3)?

+ (3 + 2?3)(1+?3)?).

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