A319970 - OEIS

%I #37 Jan 05 2025 19:51:41

%S 4,17,28,41,48,61,72,85,98,109,122,129,142,153,166,177,190,197,210,

%T 221,234,247,258,271,278,291,302,315,322,335,346,359,372,383,396,403,

%U 416,427,440,451,464,471,484,495,508,521,532,545,552,565,576,589,602,613

%N a(n) = A003146(A003144(n)).

%C 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 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

%C 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

%H Rémy Sigrist, <a href="/A319970/b319970.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 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 Y.-K. Huang, Z.-Y. Wen, <a href="https://doi.org/10.1016/S0252-9602(15)30086-2">Kernel words and gap sequence of the Tribonacci sequence</a>, Acta Mathematica Scientia (Series B). 36.1 (2016) 173-194.

%F a(n) = A003146(A003144(n)).

%F a(n) = 2*(A003144(n) + A003145(n)) + n - 3 = 2*(A278040(n-1) + A278039(n-1)) + n + 1, n >= 1. For a proof see the W. Lang link in A278040, Proposition 9, eq. (55). _Wolfdieter Lang_, Apr 11 2019

%F a(1) = 4, a(n+1) = 4 + Sum_{k=1..n} d(k), where d is the tribonacci sequence on the alphabet {13,11,7}. - _Michel Dekking_, Oct 04 2019

%Y Cf. A003144, A003145, A003146, A003622, A278039, A278040, A278041, and A319966-A319972.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Oct 05 2018

%E More terms from _Rémy Sigrist_, Oct 16 2018