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Number of 1-unbordered words of length n over a 3-letter alphabet beginning with a fixed letter.
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%I #42 Aug 16 2024 21:18:03

%S 1,1,3,7,21,57,171,499,1497,4449,13347,39927,119781,359001,1077003,

%T 3230011,9690033,29067105,87201315,261595047,784785141,2354328729,

%U 7062986187,21188878707,63566636121,190699668801,572099006403,1716296301207,5148888903621,15446664556857

%N Number of 1-unbordered words of length n over a 3-letter alphabet beginning with a fixed letter.

%C a(n) is equal to the number of short leaves in the tautological degree 3 lamination at depth n+1.

%H Danny Calegari, <a href="https://arxiv.org/abs/2106.00578">Combinatorics of the Tautological Lamination</a>, arXiv:2106.00578, [math.DS], 2021-2024.

%H Danny Calegari, <a href="https://doi.org/10.2140/pjm.2024.329.39">Combinatorics of the Tautological Lamination</a>, Pacific J. Math. 329 (2024) 39-61.

%F a(1) = 1; a(2*n+1) = 3*a(2*n) for n > 0; a(2*n) = 3*a(2*n-1) - 2*a(n) for n > 0.

%o (PARI) a(n) = if (n==1, 1, if (n%2, 3*a(n-1), 3*a(n-1)-2*a(n/2))); \\ _Michel Marcus_, Aug 09 2024

%K nonn

%O 1,3

%A _Danny Calegari_, Aug 08 2024

%E More terms from _Michel Marcus_, Aug 09 2024