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6, 19, 30, 43, 50, 63, 74, 87, 100, 111, 124, 131, 144, 155, 168, 179, 192, 199, 212, 223, 236, 249, 260, 273, 280, 293, 304, 317, 324, 337, 348, 361, 374, 385, 398, 405, 418, 429, 442, 453, 466, 473, 486, 497, 510, 523, 534, 547, 554, 567, 578, 591, 604, 615, 628, 635, 648, 659, 672, 683, 696, 703, 716, 727, 740, 753
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OFFSET
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1,1
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COMMENTS
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In another version with the tribonacci word TriWord = A080843 (written as a sequence which has offset 0) and the positions of 0, 1 and 2 given by the B = A278039, A = A278040 and C = A278041 numbers, respectively, the present sequence (with offset 0) gives the smaller of the B-number pairs (B(k), B(k+1)) with B(k+1) = B(k) + 1 for some k >= 0 (named tribonacci B0-numbers), ordered increasingly.
The B-numbers A278039 come in three disjoint and complementary types, called B0-, B1- and B2-numbers. They are defined by the indices k of pairs of consecutive entries TriWord(k), Triword(k+1) depending on their values 0, 0 or 0, 1 or 0, 2 for the B0- or B1- or B2-numbers, respectively.
The B0-numbers are a(n+1) = 2*C(n) - n = A(A(n)) + 1; the B1-numbers are B1(n) = A(n) - 1; and the B2-numbers are B2(n) = C(n) - 1, all for n >= 0.
B0(n) + 1 = B1(A(n)+1), B1(n) + 1 = A(n) and B2(n) + 1 = C(n).
(End)
(a(n)) equals the positions of the word baa 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 aa is always preceded in t by the letter b, and the formula BB = AC-1, where A := A003144, B := A003145, C := A003146. - Michel Dekking, Apr 09 2019
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LINKS
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Elena Barcucci, Luc Belanger and Srecko Brlek, On tribonacci sequences, Fib. Q., 42 (2004), 314-320. Compare page 318.
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FORMULA
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a(n+1) = B(C(n)) = B(C(n) + 1) - 1 = 2*(A(n) + B(n)) + n + 4, for n >= 0, where B = A278039, C = A278041 and A = A278040. For a proof see the W. Lang link in A278040, Proposition 9, eq. (53). - Wolfdieter Lang, Dec 13 2018
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EXAMPLE
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The TriWord A080843 starts: 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, ... (offset 0)
The trisection of the B-numbers A278039 (indices for 0 in TriWord) begins:
n : 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
B0: 6 19 30 43 50 63 74 87 100 111 124 131 144 155 168 179 192 ...
B1: 0 4 7 11 13 17 20 24 28 31 35 37 41 44 48 51 55 ...
B2: 2 9 15 22 26 33 39 46 53 59 66 70 77 83 90 96 103 ...
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(End)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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