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A307247
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Second digit in the expansion of n in Fraenkel's exotic ternary representation.
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1
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0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 1
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OFFSET
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1,7
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COMMENTS
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Let {p_i, i >= 0} = {1,3,7,17,41,99,...} denote the numerators of successive convergents to sqrt(2) (see A001333). Then any n >= 0 has a unique representation as n = Sum_{i >= 0} d_i*p_i, with 0 <= d_i <= 2, d_{i+1}=2 => d_i=0. Sequence gives a(n+1) = d_1.
Let x be the 3-symbol Pell word A294180 = 1, 2, 3, 1, 2, 3, 1, 1, 2, 3, 1, 2, ... Let delta be the morphism
1 -> 000, 2 -> 111, 3 -> 2.
Then delta(x) = (a(n)). This can be proved by induction, starting from the knowledge that the sequence of first digits d_0 = d_0(n) of n in the exotic ternary expansion shifted by 1 is equal to x (see A263844).
More generally, the sequence of k-th digits d_k shifted by 1 is equal to delta_k(x), where the morphism delta_k is given by
1 -> U_k, 2 -> V_k, 3 -> W_k.
Here U_k is a concatenation of p_{k+1} letters 0, V_k is a concatenation of p_{k+1} letters 1, and W_k is a concatenation of p_k letters 2.
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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