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It could be more appropriate to consider the sequence as a kind of two-dimensional tribonacci sequence (a(2n-1),(a(2n)), i.e. as (1, 1), (1, 2), (2, 3), (4, 6), (7, 11), (13, 20), (24, 37), (44, 68), (81, 125), (149, 230), (274, 423), (504, 778), (927, 1431), (1705, 2632), (3136, 4841),... since after the first three initial pairs, the next pair can be obtained by adding three previous pairs component-wise. However, the first three initial pairs (1, 1), (1, 2), (2, 3) are redundant in comparison with the original integer sequence that needs only three initial integers 1, 1 and 1.

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(MAGMAMagma) I:=[1, 1, 1, 2, 2, 3]; [n le 6 select I[n] else Self(n-2) + Self(n-4) + Self(n-6): n in [1..50]]; // G. C. Greubel, Nov 03 2018

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It could be more appropriate to consider the sequence as a kind of two -dimensional tribonacci sequence (a(2n-1),(a(2n)), i.e. as (1, 1), (1, 2), (2, 3), (4, 6), (7, 11), (13, 20), (24, 37), (44, 68), (81, 125), (149, 230), (274, 423), (504, 778), (927, 1431), (1705, 2632), (3136, 4841),... since after the first three initial pairs, the next pair can be obtained by adding three previous pairs component-wise. However, the first three initial pairs (1, 1), (1, 2), (2, 3) are redundant in comparison with the original integer sequence that needs only three initial integers 1, 1 and 1.

One method to construct the two -dimensional sequence is by using the well-known tribonacci-related morphism f with f(a) = ab, f(b) = ac, f(c) = a on the monoid of strings over the alphabet {a, b, c}. Using component-wise map, the following sequence of pairs is obtained: (c,b), (a, ac), (ab, aba), (abac, abacab), (abacaba, abacabaabac), (abacabaabacab, abacabaabacababacaba), ...; which is initialized by the pair (c,b) and any pair (x,y) is followed by (f(x),f(y)). The length of every string in every component consitutes the two-dimensional sequence.

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