cp's OEIS Frontend

Tribonacci numbers 4*A001590(n+2) (i.e., tribonacci numbers t(n) = t(n-1) + t(n-2) + t(n-3) with t(0) = 0, t(1) = 4, t(2) = 8) for n > 2 are numbers of ternary sequences (q(1),q(2),...,q(n)), q(i) = 0,1,2, of length n such that all triples (q(i),q(i+1),q(i+2)) contain digits 0 and 1 at least once. In the other words {0,1} is a subset of each {q(i),q(i+1),q(i+2)}. Similarly, A075092(n), for n > 2, presents a number of ternary cyclic sequences with this property. Hence, a(n), for n > 2, gives a number of (ordinary) sequences which do not lead to cyclic sequences with triples (q(n-1),q(n),q(1)) and (q(n),q(1),q(2)) satisfying the above formulated condition. The special cases, n = 1,2, can be included in a way proposed in A001644.

The recurrence formula is the same as this for A075092, with different initial conditions.

a(n) is the reflected sequence (cf. A074058) of the generalized tribonacci sequence b(n) with b(0) = 3 and b(n) = A186575(n-1) for n > 0.

A generalized tribonacci (A001590) sequence.

For n > 2, 6*a(n) is the number of quaternary sequences of length n in which all triples (q(i),q(i+1),q(i+2)) contain two (arbitrarily chosen) digits, e.g., 0 and 3.

Similarly, recurrences a(n) = a(n-1) + a(n-2) + k*a(n-3) are related to binary (k=0, the Fibonacci sequence A000045), ternary (k=1, the tribonacci sequence A001590), quinary (k=3) and so on sequences with all triples (t(i),t(i+1),t(i+2)) containing two (arbitrarily chosen) digits (usually 0 and k+1).

For n > 0, a(n) is the number of ways to tile a strip of length n with squares, dominoes, and two colors of trominoes, with the restriction that the first tile cannot be a tromino. - Greg Dresden and Bora Bursalı, Aug 31 2023

For n > 1, a(n) is the number of ways to tile a strip of length n-2 with squares, dominoes, and two colors of trominoes, where the strip begins with an upper level of two cells. For example, when n=7 we have this strip of length 5:

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|||_|||. - Guanji Chen and Greg Dresden, Jun 17 2024