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A278040 The tribonacci representation of a(n) is obtained by appending 0,1 to the tribonacci representation of n (cf. A278038).

%I A278040 #59 Jan 05 2025 19:51:41
%S A278040 1,5,8,12,14,18,21,25,29,32,36,38,42,45,49,52,56,58,62,65,69,73,76,80,
%T A278040 82,86,89,93,95,99,102,106,110,113,117,119,123,126,130,133,137,139,
%U A278040 143,146,150,154,157,161,163,167,170,174,178,181,185,187,191,194,198,201,205,207,211,214,218,222,225,229,231,235
%N A278040 The tribonacci representation of a(n) is obtained by appending 0,1 to the tribonacci representation of n (cf. A278038).
%C A278040 This sequence gives the A(n) numbers of the W. Lang link. There the B(n) and C(n) numbers are A278039(n) and A278041(n), respectively. - _Wolfdieter Lang_, Dec 05 2018
%C A278040 Positions of letter b in the tribonacci word t generated by a->ab, b->ac, c->a, when given offset 0. - _Michel Dekking_, Apr 03 2019
%C A278040 This sequence gives the positions of the word ab in the tribonacci word t. This follows from the fact that the letter b is always preceded in t by the letter a, and the formula AA = B-1, where A := A003144, B := A003145, C := A003146. - _Michel Dekking_, Apr 09 2019
%H A278040 N. J. A. Sloane, <a href="/A278040/b278040.txt">Table of n, a(n) for n = 0..20000</a>
%H A278040 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.
%H A278040 Wolfdieter Lang, <a href="https://arxiv.org/abs/1810.09787">The Tribonacci and ABC Representations of Numbers are Equivalent</a>, arXiv:1810.09787v1 [math.NT], 2018.
%F A278040 a(n) = A003145(n+1) - 1.
%F A278040 a(n) = A003144(A003144(n)). - _N. J. A. Sloane_, Oct 05 2018
%F A278040 See Theorem 13 in the Carlitz, Scoville and Hoggatt paper. - _Michel Dekking_, Mar 20 2019
%F A278040 From _Wolfdieter Lang_, Dec 13 2018: (Start)
%F A278040 This sequence gives the indices k with A080843(k) = 1, ordered increasingly with offset 0.
%F A278040 a(n) = 1 + 4*n - A319198(n-1), n >= 0, with A319198(-1) = 0.
%F A278040 a(n) = A276796(C(n)) - 1, with C(n) = A278041(n).
%F A278040 For a proof see the W. Lang link, Proposition 5, and eq. (58).
%F A278040 a(n) - 1 = B1(n), where B1-numbers are B-numbers from A278039 followed by an A-number from A278040. See a comment and example in A319968.
%F A278040 a(n) - 1 = B(B(n)) = B(B(n) + 1) - 2, for n > = 0, where B = A278039.
%F A278040 (End)
%e A278040 The tribonacci representation of 7 is 1000 (see A278038), so a(7) has tribonacci representation 100001, which is 24+1 = 25, so a(7) = 25.
%Y A278040 Cf. A003145, A276789, A276796, A278038, A278039, A278041, A319198, A319968.
%Y A278040 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.
%K A278040 nonn,base,easy
%O A278040 0,2
%A A278040 _N. J. A. Sloane_, Nov 18 2016