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Join[{1, 2, 4}, NestList[3#&, 8, 30]] (* Harvey P. Dale, May 14 2022 *)
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M. Archibald, A. Blecher, and A. Knopfmacher, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Blecher/arch14.html">Fixed Points in Compositions and Words</a>, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.
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a(n) = floor( (4*n-2)/(n+1) )*a(n-1). Without the floor function the recursion gives the Catalan numbers (A000108). - Hauke Woerpel, Oct 16 2020
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