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For n >= 2, a(n+2) is the number of ways to tile this figure of length n with squares, dominos, dominoes, and "trominostrominoes" (of length 3):
As an example, here is one of the 254 possible tilings of this figure of length 8 with squares, dominos, dominoes, and trominostrominoes:
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From Greg Dresden and Veda Garigipati, Jun 14 2022: (Start)
|_|_____|_|___|_|. - Greg Dresden and Veda Garigipati, Jun 14 2022
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Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, <a href="http://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
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For n >= 2, a(n+2) is the number of ways to tile this figure of length n with squares, dominos, and "trominos" (of length 3):
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As an example, here is one of the 254 possible tilings of this figure of length 8 with squares, dominos, and trominos:
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|_|_____|_|___|_|. - Greg Dresden and Veda Garigipati, Jun 14 2022