A319969 a(n) = A003145(A003146(n)).

13, 37, 57, 81, 94, 118, 138, 162, 186, 206, 230, 243, 267, 287, 311, 331, 355, 368, 392, 412, 436, 460, 480, 504, 517, 541, 561, 585, 598, 622, 642, 666, 690, 710, 734, 747, 771, 791, 815, 835, 859, 872, 896, 916, 940, 964, 984, 1008, 1021, 1045, 1065, 1089

Offset: 1

Author

N. J. A. Sloane, Oct 05 2018

Keywords

• nonn

Comments

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

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Elena Barcucci, Luc Belanger and Srecko Brlek, On tribonacci sequences, Fib. Q., 42 (2004), 314-320. Compare page 318.
• L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Fibonacci representations of higher order, Fib. Quart., 10 (1972), 43-69, Theorem 13.
• Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.

Crossrefs

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

Formula

a(n) = A003145(A003146(n)).

a(n) = 3*A003144(n) + 4*A003145(n) + 2*n = 4*A276040(n-1) + 3*A278039(n-1) + 2*n + 7, n >= 1. For a proof see the W. Lang link, Proposition 9, eq. (50). - Wolfdieter Lang, Apr 11 2019

Extensions

More terms from Rémy Sigrist, Oct 16 2018