A278041 -id:A278041 - cp's OEIS frontend

A322411 Compound tribonacci sequence with a(n) = A278040(A278041(n)), for n >= 0.

12, 36, 56, 80, 93, 117, 137, 161, 185, 205, 229, 242, 266, 286, 310, 330, 354, 367, 391, 411, 435, 459, 479, 503, 516, 540, 560, 584, 597, 621, 641, 665, 689, 709, 733, 746, 770, 790, 814, 834, 858, 871, 895, 915, 939, 963, 983, 1007, 1020, 1044, 1064, 1088, 1112, 1132, 1156, 1169, 1193, 1213, 1237, 1257, 1281

Offset: 0

Wolfdieter Lang, Jan 02 2019

The nine sequences A308199, A319967, A319968, A322410, A322409, A322411, A322413, A322412, A322414 are based on defining the tribonacci ternary word to start with index 0 (in contrast to the usual definition, in A080843 and A092782, which starts with index 1). As a result these nine sequences differ from the compound tribonacci sequences defined in A278040, A278041, and A319966-A319972. - N. J. A. Sloane, Apr 05 2019

L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Fibonacci representations of higher order, Fib. Quart., 10 (1972), 43-69.

Cf. A278040, A278041, A322410.

a(n) = A(C(n)) = A(C(n) + 1) - 2 = 4*A(n) + 3*B(n) + 2*n + 8, for n >= 0, with A = A278040 and C = A278041. For a proof see the W. Lang link in A278040, Proposition 9, eq. (50).
This formula already follows from Theorem 15 in the 1972 paper by Carlitz et al., which gives that b(c(n)) = a(n) + 2b(n) + 2c(n), where a, b and c are the classical positional sequences of the letters in the tribonacci word. The connection is made by using that c(n) = a(n) + b(n) + n, and by making the translation B(n) = a(n+1)-1, A(n) = b(n+1)-1, C(n) = c(n+1)-1. (Note the switching of A and B!). - Michel Dekking, Apr 07 2019
a(n+1) = A319969(n)-1 = A003145(A003146(n))-1, the corresponding classical compound tribonacci sequence. - Michel Dekking, Apr 04 2019

A322412 Compound tribonacci sequence with a(n) = A278041(A278040(n)), for n >= 0.

Original entry on oeis.org

10, 34, 54, 78, 91, 115, 135, 159, 183, 203, 227, 240, 264, 284, 308, 328, 352, 365, 389, 409, 433, 457, 477, 501, 514, 538, 558, 582, 595, 619, 639, 663, 687, 707, 731, 744, 768, 788, 812, 832, 856, 869, 893, 913, 937, 961, 981, 1005, 1018, 1042, 1062, 1086, 1110, 1130, 1154, 1167, 1191, 1211, 1235

Offset: 0

Wolfdieter Lang, Jan 02 2019

(a(n+1)) = A319971(n)-1 = A003146(A003145(n))-1, the corresponding classical compound tribonacci sequence. - Michel Dekking

The nine sequences A308199, A319967, A319968, A322410, A322409, A322411, A322413, A322412, A322414 are based on defining the tribonacci ternary word to start with index 0 (in contrast to the usual definition, in A080843 and A092782, which starts with index 1). As a result these nine sequences differ from the compound tribonacci sequences defined in A278040, A278041, and A319966-A319972. - N. J. A. Sloane, Apr 05 2019

Cf. A278040, A278041, A322411.

a(n) = C(A(n)) = C(A(n) + 1) - 6 = 4*A(n) + 3*B(n) + 2*(n+3). for n >= 0, where A = A278040, B = A278039 and C = A278041. For a proof see the W. Lang link in A278040, Proposition 9, eq. (54).

A322413 Compound tribonacci sequence with a(n) = A278041(A278039(n)), for n >= 0.

Original entry on oeis.org

3, 16, 27, 40, 47, 60, 71, 84, 97, 108, 121, 128, 141, 152, 165, 176, 189, 196, 209, 220, 233, 246, 257, 270, 277, 290, 301, 314, 321, 334, 345, 358, 371, 382, 395, 402, 415, 426, 439, 450, 463, 470, 483, 494, 507, 520, 531, 544, 551, 564, 575, 588, 601, 612, 625, 632, 645, 656, 669, 680, 693

Offset: 0

Wolfdieter Lang, Jan 02 2019

(a(n+1)) = A319970(n)-1 = A003146(A003144(n))-1, the corresponding classical compound tribonacci sequence. - Michel Dekking, Apr 03 2019

The nine sequences A308199, A319967, A319968, A322410, A322409, A322411, A322413, A322412, A322414 are based on defining the tribonacci ternary word to start with index 0 (in contrast to the usual definition, in A080843 and A092782, which starts with index 1). As a result these nine sequences differ from the compound tribonacci sequences defined in A278040, A278041, and A319966-A319972. - N. J. A. Sloane, Apr 05 2019

Cf. A278039, A278040, A278041, A322412.

Cf. A003144, A003145, A003146.

a(n) = C(B(n)) = C(B(n) + 1) - 7 = 2*(A(n) + B(n)) + n + 1, for n >= 0, where A = A278040, B = A278039 and C = A278041. For a proof see the W. Lang link in A278040, Proposition 9, eq. (55).

A322414 Compound tribonacci sequence with a(n) = A278041(A278041(n)), for n >= 0.

Original entry on oeis.org

23, 67, 104, 148, 172, 216, 253, 297, 341, 378, 422, 446, 490, 527, 571, 608, 652, 676, 720, 757, 801, 845, 882, 926, 950, 994, 1031, 1075, 1099, 1143, 1180, 1224, 1268, 1305, 1349, 1373, 1417, 1454, 1498, 1535, 1579, 1603, 1647, 1684, 1728, 1772, 1809, 1853, 1877, 1921, 1958, 2002, 2046, 2083, 2127, 2151, 2195, 2232, 2276, 2313, 2357

Offset: 0

Wolfdieter Lang, Jan 02 2019

(a(n+1)) = A319972(n)-1 = A003146(A003146(n))-1, the corresponding classical compound tribonacci sequence. - Michel Dekking, Apr 04 2019

The nine sequences A308199, A319967, A319968, A322410, A322409, A322411, A322413, A322412, A322414 are based on defining the tribonacci ternary word to start with index 0 (in contrast to the usual definition, in A080843 and A092782, which starts with index 1). As a result these nine sequences differ from the compound tribonacci sequences defined in A278040, A278041, and A319966-A319972. - N. J. A. Sloane, Apr 05 2019

Cf. A278039, A278040, A278041, A322413.

Cf. A003144, A003145, A003146.

a(n) = C(C(n)) = C(C(n) + 1) - 4 = 7*A(n) + 6*B(n) + 4*(n + 4), for n >= 0, where A = A278040, B = A278039 and C = A278041. For a proof see the W. Lang link in A278040, Proposition 9, eq. (56).

A316717 a(n) is the number of 3s in A316713(n). That is, a(n) is the number of C-sequences (A278041) used in the tribonacci ABC-representation of n >= 0.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1

Offset: 0

Wolfdieter Lang, Sep 11 2018

nonn

The number of 1s and 2s in A316713(n) is given in A316715 and A316716, respectively.

			See column #(3) in A316713.

Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.

Cf. A316713, A316714, A316715, A316716.

a(n) = number of 3s in A316713(n), that is the number of Cs in the tribonacci ABC-representation of n >= 0.

A322408 Compound sequence with a(n) = A319198(A278041(n)), for n >= 0.

Original entry on oeis.org

3, 7, 11, 15, 18, 22, 26, 30, 34, 38, 42, 45, 49, 53, 57, 61, 65, 68, 72, 76, 80, 84, 88, 92, 95, 99, 103, 107, 110, 114, 118, 122, 126, 130, 134, 137, 141, 145, 149, 153, 157, 160, 164, 168, 172, 176, 180, 184, 187, 191, 195, 199, 203, 207, 211, 214, 218, 222, 226, 230, 234

Offset: 0

Wolfdieter Lang, Jan 02 2019

Old name was: Compound tribonacci sequence a(n) = A319198(A278041(n)), for n >= 0.
a(n) gives the sum of the entries of the tribonacci word sequence t = A080843 not exceeding t(C(n)), with C(n) = A278041(n).
The nine sequences A308199, A319967, A319968, A322410, A322409, A322411, A322413, A322412, A322414 are based on defining the tribonacci ternary word to start with index 0 (in contrast to the usual definition, in A080843 and A092782, which starts with index 1). As a result these nine sequences differ from the compound tribonacci sequences defined in  A278040, A278041, and A319966-A319972. - N. J. A. Sloane, Apr 05 2019
The difference sequence (a(n+1)-a(n)) is equal to a change of alphabet of the tribonacci word t = A092782. The alphabet is {4,4,3}. This follows from the formula a(n) = A278039(n) + 2*n + 3. - Michel Dekking, Oct 05 2019

			n = 2: C(2) = 16, t = {0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, ...} which sums to 11 = a(2) = 4 + 7, because B(2) = 4.

Cf. A080843, A278039, A278041, A319198, A321333, A322407.

a(n) =  z(C(n)) = Sum_{j=0..C(n)} t(j), n >= 0, with z = A319198, C = A278041 and t = A080843.
a(n) = B(n) + 2*n + 3, where B(n) = A278039(n). For a proof see the W. Lang link in A080843, Proposition 8, eq. (47).
a(n) = 3 + Sum_{k=1..n-1} d(k), where d is the tribonacci sequence on the alphabet {4,4,3}. - Michel Dekking, Oct 05 2019

Name changed by Michel Dekking, Oct 08 2019

A003146 Positions of letter c in the tribonacci word abacabaabacababac... generated by a->ab, b->ac, c->a (cf. A092782).

Original entry on oeis.org

4, 11, 17, 24, 28, 35, 41, 48, 55, 61, 68, 72, 79, 85, 92, 98, 105, 109, 116, 122, 129, 136, 142, 149, 153, 160, 166, 173, 177, 184, 190, 197, 204, 210, 217, 221, 228, 234, 241, 247, 254, 258, 265, 271, 278, 285, 291, 298, 302, 309, 315, 322, 329, 335, 342, 346, 353, 359

Offset: 1

N. J. A. Sloane

nonn

Comment from Philippe Deléham, Feb 27 2009: A003144, A003145, A003146 may be defined as follows. Consider the map psi: a -> ab, b -> ac, c -> a. The image (or trajectory) of a under repeated application of this map is the infinite word a, b, a, c, a, b, a, a, b, a, c, a, b, a, b, a, c, ... (setting a = 1, b = 2, c = 3 gives A092782). The indices of a, b, c give respectively A003144, A003145, A003146.
The infinite word may also be defined as the limit S_oo where S_1 = a, S_n = psi(S_{n-1}). Or, by S_1 = a, S_2 = ab, S_3 = abac, and thereafter S_n = S_{n-1} S_{n-2} S_{n-3}. It is the unique word such that S_oo = psi(S_oo).
Also, indices of c in the sequence closed under a -> abac, b -> aba, c -> ab; starting with a(1) = a. - Philippe Deléham, Apr 16 2004
Theorem: A number m is in this sequence iff the tribonacci representation of m-1 ends with 11. [Duchene and Rigo, Remark 2.5] - N. J. A. Sloane, Mar 02 2019

Eric Duchêne, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, Urban Larsson, Wythoff Visions, Games of No Chance, Vol. 5; MSRI Publications, Vol. 70 (2017), pages 101-153.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..10609
Elena Barcucci, Luc Belanger and Srecko Brlek, On tribonacci sequences, Fib. Q., 42 (2004), 314-320.
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Fibonacci representations of higher order, Fib. Quart., 10 (1972), 43-69. The present sequence is called c.
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
Eric Duchêne and Michel Rigo, A morphic approach to combinatorial games: the Tribonacci case. RAIRO - Theoretical Informatics and Applications, 42, 2008, pp 375-393. doi:10.1051/ita:2007039. [Also available from Numdam]
A. J. Hildebrand, Junxian Li, Xiaomin Li, Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018.
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.

Cf. A003145, A003144, A080843, A092782, A058265, A276791, A276798, A276801, A277721.
First differences are A276792. A278041 (subtract 1 from each term, and use offset 0).
For tribonacci representations of numbers see A278038.

M:=17; S[1]:=`a`; S[2]:=`ab`; S[3]:=`abac`;
for n from 4 to M do S[n]:=cat(S[n-1], S[n-2], S[n-3]); od:
t0:=S[M]: l:=length(t0); t1:=[];
for i from 1 to l do if substring(t0,i..i) = `c` then t1:=[op(t1),i]; fi; od:
# N. J. A. Sloane, Nov 01 2006

StringPosition[SubstitutionSystem[{"a" -> "ab", "b" -> "ac", "c" -> "a"}, "c", {#}][[1]], "c"][[All, 1]] &@ 11 (* Michael De Vlieger, Mar 30 2017, Version 10.2, after JungHwan Min at A003144 *)

It appears that a(n) = floor(n*t^3) + eps for all n, where t is the tribonacci constant A058265 and eps is 0, 1, 2, or 3. See A277721. - N. J. A. Sloane, Oct 28 2016. This is true - see the Dekking et al. paper. - N. J. A. Sloane, Jul 22 2019

More terms from Philippe Deléham, Apr 16 2004

Entry revised by N. J. A. Sloane, Oct 13 2016

A278040 The tribonacci representation of a(n) is obtained by appending 0,1 to the tribonacci representation of n (cf. A278038).

Original entry on oeis.org

1, 5, 8, 12, 14, 18, 21, 25, 29, 32, 36, 38, 42, 45, 49, 52, 56, 58, 62, 65, 69, 73, 76, 80, 82, 86, 89, 93, 95, 99, 102, 106, 110, 113, 117, 119, 123, 126, 130, 133, 137, 139, 143, 146, 150, 154, 157, 161, 163, 167, 170, 174, 178, 181, 185, 187, 191, 194, 198, 201, 205, 207, 211, 214, 218, 222, 225, 229, 231, 235

Offset: 0

N. J. A. Sloane, Nov 18 2016

This sequence gives the A(n) numbers of the W. Lang link. There the B(n) and C(n) numbers are A278039(n) and A278041(n), respectively. - Wolfdieter Lang, Dec 05 2018
Positions of letter b 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 ab in the tribonacci word t. This follows from the fact that the letter b is always preceded in t by the letter a, and the formula AA = B-1, where A := A003144, B := A003145, C := A003146. - Michel Dekking, Apr 09 2019

			The tribonacci representation of 7 is 1000 (see A278038), so a(7) has tribonacci representation 100001, which is 24+1 = 25, so a(7) = 25.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Fibonacci representations of higher order, Fib. Quart., 10 (1972), 43-69.
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv:1810.09787v1 [math.NT], 2018.

Cf. A003145, A276789, A276796, A278038, A278039, A278041, A319198, A319968.

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.

a(n) = A003145(n+1) - 1.
a(n) = A003144(A003144(n)). - N. J. A. Sloane, Oct 05 2018
See Theorem 13 in the Carlitz, Scoville and Hoggatt paper. - Michel Dekking, Mar 20 2019
From Wolfdieter Lang, Dec 13 2018: (Start)
This sequence gives the indices k with A080843(k) = 1, ordered increasingly with offset 0.
a(n) = 1 + 4*n - A319198(n-1), n >= 0, with A319198(-1) = 0.
a(n) = A276796(C(n)) - 1, with C(n) = A278041(n).
For a proof see the W. Lang link, Proposition 5, and eq. (58).
a(n) - 1 = B1(n), where B1-numbers are B-numbers from A278039 followed by an A-number from A278040. See a comment and example in A319968.
a(n) - 1 = B(B(n)) = B(B(n) + 1) - 2, for n > = 0, where B = A278039.
(End)

A278039 The tribonacci representation of a(n) is obtained by appending a 0 to the tribonacci representation of n (cf. A278038).

Original entry on oeis.org

0, 2, 4, 6, 7, 9, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 30, 31, 33, 35, 37, 39, 41, 43, 44, 46, 48, 50, 51, 53, 55, 57, 59, 61, 63, 64, 66, 68, 70, 72, 74, 75, 77, 79, 81, 83, 85, 87, 88, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 111, 112, 114, 116, 118, 120, 122, 124, 125, 127, 129, 131, 132, 134, 136

Offset: 0

N. J. A. Sloane, Nov 18 2016

This sequence records the indices for the 0 values of A080843, ordered increasingly. In the W. Lang link a(n) = B(n). - Wolfdieter Lang, Dec 06 2018

Sequence gives the positions of letter a in the tribonacci word generated by a->ab, b->ac, c->a, when given offset 0. - Michel Dekking, Apr 03 2019

			The tribonacci representation of 7 is 1000 (see A278038), so a(7) has tribonacci representation 10000, which is 13, so a(7) = 13.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv:1810.09787v1 [math.NT], 2018.
Bo Tan and Zhi-Ying Wen, Some properties of the Tribonacci sequence, European Journal of Combinatorics, 28 (2007) 1703-1719. See the sequence O_1(n).

Cf. A003144, A276798, A278038, A278040, A278041.

Partial sums of A276788.

a(n) = A003144(n+1) - 1 = Sum_{k=1..n} A276788(k), n >= 0 (an empty sum is 0).

a(n) = 2*n - (A276798(n) - 1), n >= 0. For a proof see the link, Proposition 6 B). - Wolfdieter Lang, Dec 04 2018

A319966 a(n) = A003144(A003146(n)).

Original entry on oeis.org

7, 20, 31, 44, 51, 64, 75, 88, 101, 112, 125, 132, 145, 156, 169, 180, 193, 200, 213, 224, 237, 250, 261, 274, 281, 294, 305, 318, 325, 338, 349, 362, 375, 386, 399, 406, 419, 430, 443, 454, 467, 474, 487, 498, 511, 524, 535, 548, 555, 568, 579, 592, 605, 616

Offset: 1

N. J. A. Sloane, Oct 05 2018

nonn

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 aa 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 aa, ab and ac give a splitting of the positional sequence of the letter a, and the three sets AA(N), AB(N) and AC(N), give a splitting of the set A(N). Here A := A003144, B := A003145, C := A003146, and N is the set of positive integers. - Michel Dekking, Apr 09 2019

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.
Rémy Sigrist, Perl program for A319966

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

Perl
```
See Links section.
```

a(n) = A319968(n) + 1. - Michel Dekking, Apr 04 2019

More terms from Rémy Sigrist, Oct 16 2018

cp's OEIS Frontend

A322411 Compound tribonacci sequence with a(n) = A278040(A278041(n)), for n >= 0.

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A322412 Compound tribonacci sequence with a(n) = A278041(A278040(n)), for n >= 0.

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A322413 Compound tribonacci sequence with a(n) = A278041(A278039(n)), for n >= 0.

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A322414 Compound tribonacci sequence with a(n) = A278041(A278041(n)), for n >= 0.

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A316717 a(n) is the number of 3s in A316713(n). That is, a(n) is the number of C-sequences (A278041) used in the tribonacci ABC-representation of n >= 0.

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A322408 Compound sequence with a(n) = A319198(A278041(n)), for n >= 0.

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A003146 Positions of letter c in the tribonacci word abacabaabacababac... generated by a->ab, b->ac, c->a (cf. A092782).

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A278040 The tribonacci representation of a(n) is obtained by appending 0,1 to the tribonacci representation of n (cf. A278038).

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A278039 The tribonacci representation of a(n) is obtained by appending a 0 to the tribonacci representation of n (cf. A278038).

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A319966 a(n) = A003144(A003146(n)).