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A319968 a(n) = A003145(A003145(n)).

%I A319968 #51 Jan 05 2025 19:51:41
%S A319968 6,19,30,43,50,63,74,87,100,111,124,131,144,155,168,179,192,199,212,
%T A319968 223,236,249,260,273,280,293,304,317,324,337,348,361,374,385,398,405,
%U A319968 418,429,442,453,466,473,486,497,510,523,534,547,554,567,578,591,604,615,628,635,648,659,672,683,696,703,716,727,740,753
%N A319968 a(n) = A003145(A003145(n)).
%C A319968 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.
%C A319968 From _Wolfdieter Lang_, Oct 19 2018: (Start)
%C A319968 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.
%C A319968 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.
%C A319968 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.
%C A319968 B0(n) + 1 = B1(A(n)+1),  B1(n) + 1 = A(n) and B2(n) + 1 = C(n).
%C A319968 (End)
%C A319968 (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
%H A319968 Rémy Sigrist, <a href="/A319968/b319968.txt">Table of n, a(n) for n = 1..10000</a>
%H A319968 Elena Barcucci, Luc Belanger and Srecko Brlek, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/42-4/quartbarcucci04_2004.pdf">On tribonacci sequences</a>, Fib. Q., 42 (2004), 314-320. Compare page 318.
%H A319968 L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/10-1/carlitz3-a.pdf">Fibonacci representations of higher order</a>, Fib. Quart., 10 (1972), 43-69, Theorem 13.
%H A319968 Wolfdieter Lang, <a href="https://arxiv.org/abs/1810.09787">The Tribonacci and ABC Representations of Numbers are Equivalent</a>, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
%F A319968 a(n) = A003145(A003145(n)), for n >= 1.
%F A319968 a(n) = B0(n-1) = 2*A003146(n) - (n+1) = 2*A278041(n-1) - (n-1) = A278040(A278040(n-1)) + 1, for n >= 1. For B0 see a comment above and the example. - _Wolfdieter Lang_, Oct 19 2018
%F A319968 a(n+1) = B(C(n)) = B(C(n) + 1) - 1 = 2*(A(n) + B(n)) + n + 4, for n >= 0, where B = A278039, C = A278041 and A = A278040. For a proof see the W. Lang link in A278040, Proposition 9, eq. (53). - _Wolfdieter Lang_, Dec 13 2018
%F A319968 a(n) = 2*(A003144(n) + A003145(n)) + n - 1, n >= 1. [Rewriting a formula  of the precedimg entry]. - _Wolfdieter Lang_, Apr 11 2019
%e A319968 From _Wolfdieter Lang_, Oct 19 2018: (Start)
%e A319968 The TriWord A080843 starts: 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, ... (offset 0)
%e A319968 The trisection of the B-numbers A278039 (indices for 0 in TriWord) begins:
%e A319968 n :  0   1   2   3   4   5   6   7    8    9   10   11   12   13   14   15   16 ...
%e A319968 B0:  6  19  30  43  50  63  74  87  100  111  124  131  144  155  168  179  192 ...
%e A319968 B1:  0   4   7  11  13  17  20  24   28   31   35   37   41   44   48   51   55 ...
%e A319968 B2:  2   9  15  22  26  33  39  46   53   59   66   70   77   83   90   96  103 ...
%e A319968 ------------------------------------------------------------------------------------
%e A319968 (End)
%Y A319968 Cf. A003144, A003145, A003146, A003622, A080843, A278039, A278040, A278041, and A319966-A319972.
%K A319968 nonn,easy
%O A319968 1,1
%A A319968 _N. J. A. Sloane_, Oct 05 2018
%E A319968 More terms from _Joerg Arndt_, Oct 15 2018