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A278041 The tribonacci representation of a(n) is obtained by appending 0,1,1 to the tribonacci representation of n (cf. A278038).

%I A278041 #53 Jan 05 2025 19:51:41
%S A278041 3,10,16,23,27,34,40,47,54,60,67,71,78,84,91,97,104,108,115,121,128,
%T A278041 135,141,148,152,159,165,172,176,183,189,196,203,209,216,220,227,233,
%U A278041 240,246,253,257,264,270,277,284,290,297,301,308,314,321,328,334,341,345,352,358,365,371,378,382,389,395,402,409,415
%N A278041 The tribonacci representation of a(n) is obtained by appending 0,1,1 to the tribonacci representation of n (cf. A278038).
%C A278041 This sequence gives the indices k for which A080843(k) = 2, sorted increasingly with offset 0. In the W. Lang link a(n) = C(n). - _Wolfdieter Lang_, Dec 06 2018
%C A278041 Positions of letter c in the tribonacci word t generated by a->ab, b->ac, c->a, when given offset 0. - _Michel Dekking_, Apr 03 2019
%C A278041 This sequence gives the positions of the word ac in the tribonacci word t.  This follows from the fact that the letter c is always preceded in t by the letter a, and the formula AB = C-1, where A := A003144, B := A003145, C := A003146. - _Michel Dekking_, Apr 09 2019
%H A278041 N. J. A. Sloane, <a href="/A278041/b278041.txt">Table of n, a(n) for n = 0..20000</a>
%H A278041 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 A278041 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 A278041 a(n) = A003146(n+1) - 1.
%F A278041 a(n) = A003144(A003145(n)). - _N. J. A. Sloane_, Oct 05 2018
%F A278041 From _Wolfdieter Lang_, Dec 06 2018: (Start)
%F A278041 a(n) = n + 2 + A(n) + B(n), where A(n) = A278040(n) and B = A278039(n).
%F A278041 a(n) = 7*n + 3 - (z_A(n-1) + 3*z_C(n-1)), where z_A(n) = A276797(n+1) and z_C(n) = A276798(n+1) - 1, n >= 0.
%F A278041 For proofs see the W. Lang link in A080843, eqs. 37 and 40.
%F A278041 a(n) - 1 = B2(n), where B2-numbers are B-numbers from A278039 followed by a C-number from A278041. See a comment and example in A319968.
%F A278041 (End)
%e A278041 The tribonacci representation of 7 is 1000 (see A278038), so a(7) has tribonacci representation 1000011, which is 44+2+1 = 47, so a(7) = 47.
%Y A278041 Cf. A003145, A003146, A080843, A276797, A276798, A278038, A278039, A278040, A278041, A319968.
%Y A278041 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 A278041 nonn,base,easy
%O A278041 0,1
%A A278041 _N. J. A. Sloane_, Nov 18 2016