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By analogy with the Wythoff compound sequences A003622 etc., the nine compounds of A003144, A003145, A003146 might be called the tribonacci compound sequences. They are A278040, A278041, and A319966-A319972.

From Wolfdieter Lang, Oct 19 2018: (Start)

In another version with the tribonacci word TriWord = A080843 (written as a sequence which has offset 0) and the positions of 0, 1 and 2 given by the B = A278039, A = A278040 and C = A278041 numbers, respectively, the present sequence (with offset 0) gives the smaller of the B-number pairs (B(k), B(k+1)) with B(k+1) = B(k) + 1 for some k >= 0 (named tribonacci B0-numbers), ordered increasingly.

The B-numbers A278039 come in three disjoint and complementary types, called B0-, B1- and B2-numbers. They are defined by the indices k of pairs of consecutive entries TriWord(k), Triword(k+1) depending on their values 0, 0 or 0, 1 or 0, 2 for the B0- or B1- or B2-numbers, respectively.

The B0-numbers are a(n+1) = 2*C(n) - n = A(A(n)) + 1; the B1-numbers are B1(n) = A(n) - 1; and the B2-numbers are B2(n) = C(n) - 1, all for n >= 0.

B0(n) + 1 = B1(A(n)+1), B1(n) + 1 = A(n) and B2(n) + 1 = C(n).

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(a(n)) equals the positions of the word baa in the tribonacci word t = abacabaa..., fixed point of the morphism a->ab, b->ac, c->a. This follows from the fact that the word aa is always preceded in t by the letter b, and the formula BB = AC-1, where A := A003144, B := A003145, C := A003146. - Michel Dekking, Apr 09 2019

The lower and upper Wythoff sequences, A and B, satisfy the complementary equation BAA = A+2B-3.

Also numbers with suffix string 0010, when written in Zeckendorf representation (with leading zero's for the first term). - A.H.M. Smeets, Mar 20 2024

The asymptotic density of this sequence is 1/phi^4 = A094214^4 = 0.145898... . - Amiram Eldar, Mar 24 2025

The lower and upper Wythoff sequences, A and B, satisfy the complementary equation ABB = 2A+3B.

The asymptotic density of this sequence is 1/phi^5 = phi^5 - 11 = A244593 - 4 = 0.0901699... . - Amiram Eldar, Mar 24 2025

The lower and upper Wythoff sequences, A and B, satisfy the complementary equation BBB = 3A+5B.

The asymptotic density of this sequence is 1/phi^6 = A094214^6 = 0.05572809... . - Amiram Eldar, Mar 24 2025

Let tau = (1+sqrt(5))/2; then the missing numbers 3,5,8,10,13,16,18,21,... are given by round(tau^2*k) for k > 0 (A004937).

Indices n such that a(n) = a(n+1) are given by floor(tau^2*k) - 1 for k > 0 (A003622).

Numbers n such that a(n) differs from a(n+1) are given by floor(tau*k+1/tau) for k > 0 (A022342).

Indices n giving isolated terms (a(n) differs from a(n-1) and a(n+1)) are given by floor(tau*floor(tau^2*k)) for k > 0 (A003623).

Remove 0's from the first differences of sorted values; then you get a version of the infinite Fibonacci word (A001468). I.e., sorted values are 1,1,2,4,4,6,7,7,9,9,11,12,12,..., first differences are 0,1,2,0,2,1,0,2,0,2,1,0,2,0,1,...; removing 0's gives 1,2,2,1,2,2,1,2,1,2,2,1,2,1,2,2,1,2,... #{ k : a(k)=k}=infty.

The lower and upper Wythoff sequences, A and B, satisfy the complementary equation BAB = 2A+3B-1.

Also numbers with suffix string 1010, when written in Zeckendorf representation. - A.H.M. Smeets, Mar 24 2024

The asymptotic density of this sequence is 1/phi^5 = phi^5 - 11 = A244593 - 4 = 0.0901699... . - Amiram Eldar, Mar 24 2025

The history of 1-D CA Rule 90 starting from the seed pattern 1 interpreted as Zeckendorffian expansion.

Also, product of distinct terms of A001566 and appropriate Fibonacci or Lucas numbers: a(n) = FL(n+2)Product(L(2^i)^bit(n,i),i=0..) Here L(2^i) = A001566 and FL(n) = n-th Fibonacci number if n even, n-th Lucas number if n odd. bit(n,i) is the i-th digit (0 or 1) in the binary expansion of n, with the least significant digit being bit(n,0).

From Peter Munn, Apr 28 2025: (Start)

Each row in the Wythoff array, A035513, and this extended array satisfies the Fibonacci recurrence; that is each term after the first 2 is the sum of the preceding 2 terms.

We use F_i to denote the i-th Fibonacci term, A000045(i). In particular, we refer below to F_0 = 0, F_1 = 1 and F_2 = 1 several times. Note that to fully understand the description of the relationship between neighboring columns it is important to distinguish F_1 and F_2, although they have the same integer value. Similarly, the identity of an array term should be understood here as including its position in the array, not only its integer value.

The terms of this extended Wythoff array map 1:1 onto the nonempty finite subsets of Fibonacci terms (from F_0 onwards) that do not include both F_i and F_{i+1} for any i. With this map each term is the sum of its subset image. See the table in the examples.

Full description of the mapping with its relationship to A035513:

The (unextended) Wythoff array A035513 includes every positive integer exactly once. So, using the Zeckendorf representation (see link below), the array terms map 1:1 to nonempty finite subsets of the Fibonacci terms from F_2 onwards -- more precisely, onto those that do not include both F_i and F_{i+1} for any i. (Again, each array term is the sum of the Fibonacci numbers from the relevant subset.)

As shown in the Kimberling 1995 link, when we proceed from one term to the next in a row, the indices of the Fibonacci terms in the corresponding subset are incremented. When we proceed leftwards, the indices are decremented, with the subsets for the leftmost column being those that include F_2.

And when we add 2 columns on the left of the Wythoff array, the mapping continues to decrement the indices, so the corresponding extra subsets have F_0 (new leftmost column) or F_1 as their first Fibonacci term.

Thus the terms of this extended Wythoff array map 1:1 onto the nonempty finite subsets of Fibonacci terms (from F_0 onwards) that do not include both F_i and F_{i+1} for any i. The leftmost column is the nonnegative integers: if we were to remove F_0 (value 0) from the subset for an integer in this column, the subset would form the Zeckendorf representation of the integer, as subsets do in the unextended array.

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