cp's OEIS Frontend

(a(n+1)) = A319972(n)-1 = A003146(A003146(n))-1, the corresponding classical compound tribonacci sequence. - Michel Dekking, Apr 04 2019

The nine sequences A308199, A319967, A319968, A322410, A322409, A322411, A322413, A322412, A322414 are based on defining the tribonacci ternary word to start with index 0 (in contrast to the usual definition, in A080843 and A092782, which starts with index 1). As a result these nine sequences differ from the compound tribonacci sequences defined in A278040, A278041, and A319966-A319972. - N. J. A. Sloane, Apr 05 2019

Old name was: Compound tribonacci sequence a(n) = A319198(A278041(n)), for n >= 0.

a(n) gives the sum of the entries of the tribonacci word sequence t = A080843 not exceeding t(C(n)), with C(n) = A278041(n).

The nine sequences A308199, A319967, A319968, A322410, A322409, A322411, A322413, A322412, A322414 are based on defining the tribonacci ternary word to start with index 0 (in contrast to the usual definition, in A080843 and A092782, which starts with index 1). As a result these nine sequences differ from the compound tribonacci sequences defined in A278040, A278041, and A319966-A319972. - N. J. A. Sloane, Apr 05 2019

The difference sequence (a(n+1)-a(n)) is equal to a change of alphabet of the tribonacci word t = A092782. The alphabet is {4,4,3}. This follows from the formula a(n) = A278039(n) + 2*n + 3. - Michel Dekking, Oct 05 2019

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.

This sequence gives the positions of the word bab in the tribonacci word t = abacabaa..., fixed point of the morphism a->ab, b->ac, c->a. This follows from the fact that the positional sequences of baa, bab and bac give a splitting of the positional sequence of the word ba, and the three sets BA(N), BB(N) and BC(N), give a splitting of the set B(N), where A := A003144, B := A003145, C := A003146. Here N denotes the set of positive integers. - Michel Dekking, Apr 09 2019

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.

This sequence gives the positions of the word cabaa 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 baa is always preceded in t by the word ca, and the formula CA = BB-2, where A := A003144, B := A003145, C := A003146. See A319968 for BB. - Michel Dekking, Apr 09 2019

The fact that this sequence is the positional sequence of cabaa in the tribonacci word permits to apply Theorem 5.1. in the paper by Huang and Wen. This gives that the sequence (a(n+1)-a(n)) equals the tribonacci word on the alphabet {a(2)-a(1), a(3)-a(2), a(5)-a(4)} = {13, 11, 7}. - Michel Dekking, Oct 04 2019

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.

This sequence gives the positions of the word cabab 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 bab is always preceded in t by the word ca, and the formula CB = BC-2, where A := A003144, B := A003145, C := A003146. See A319969 for BC, the positional sequence of the word bab. - Michel Dekking, Apr 09 2019