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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).
26
3, 10, 16, 23, 27, 34, 40, 47, 54, 60, 67, 71, 78, 84, 91, 97, 104, 108, 115, 121, 128, 135, 141, 148, 152, 159, 165, 172, 176, 183, 189, 196, 203, 209, 216, 220, 227, 233, 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
OFFSET
0,1
COMMENTS
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
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
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
LINKS
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.
FORMULA
a(n) = A003146(n+1) - 1.
a(n) = A003144(A003145(n)). - N. J. A. Sloane, Oct 05 2018
From Wolfdieter Lang, Dec 06 2018: (Start)
a(n) = n + 2 + A(n) + B(n), where A(n) = A278040(n) and B = A278039(n).
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.
For proofs see the W. Lang link in A080843, eqs. 37 and 40.
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.
(End)
EXAMPLE
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.
CROSSREFS
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.
Sequence in context: A093516 A246302 A358792 * A063209 A063109 A083684
KEYWORD
nonn,base,easy
AUTHOR
N. J. A. Sloane, Nov 18 2016
STATUS
approved