A319972 -id:A319972 - cp's OEIS frontend

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

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)

A278041 The tribonacci representation of a(n) is obtained by appending 0,1,1 to the tribonacci representation of n (cf. A278038).

Original entry on oeis.org

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

N. J. A. Sloane, Nov 18 2016

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

			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.

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, Theorem 13.
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.

Cf. A003145, A003146, A080843, A276797, A276798, A278038, A278039, A278040, A278041, 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) = 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)

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

A319968 a(n) = A003145(A003145(n)).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 05 2018

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.
From Wolfdieter Lang, Oct 19 2018: (Start)
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

			From _Wolfdieter Lang_, Oct 19 2018: (Start)
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 ...
------------------------------------------------------------------------------------
(End)

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.

Cf. A003144, A003145, A003146, A003622, A080843, A278039, A278040, A278041, and A319966-A319972.

a(n) = A003145(A003145(n)), for n >= 1.
a(n) = B0(n-1) = 2*A003146(n) - (n+1) = 2*A278041(n-1) - (n-1) = A278040(A278040(n-1)) + 1, for n >= 1. For B0 see a comment above and the example. - Wolfdieter Lang, Oct 19 2018
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
a(n) = 2*(A003144(n) + A003145(n)) + n - 1, n >= 1. [Rewriting a formula  of the precedimg entry]. - Wolfdieter Lang, Apr 11 2019

More terms from Joerg Arndt, Oct 15 2018

A319967 a(n) = A003145(A003144(n)) where A003144 and A003145 are positions of '1' and '2' in the tribonacci word A092782.

Original entry on oeis.org

2, 9, 15, 22, 26, 33, 39, 46, 53, 59, 66, 70, 77, 83, 90, 96, 103, 107, 114, 120, 127, 134, 140, 147, 151, 158, 164, 171, 175, 182, 188, 195, 202, 208, 215, 219, 226, 232, 239, 245, 252, 256, 263, 269, 276, 283, 289, 296, 300, 307, 313, 320, 327, 333, 340, 344

Offset: 1

N. J. A. Sloane, Oct 05 2018

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 bac 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 ac is always preceded in t by the letter b, and the formula BA = C-2, where A := A003144, B := A003145, C := A003146. - 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.

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

Cf. A092782 (ternary tribonacci word).

a(n+1) = B(A(n)) = B(A(n) + 1) - 2 = A(n) + B(n) + n + 1, for n >= 0, where B = A278039 and A = A278040. For a proof see the W. Lang link in A278040, Proposition 9, eq. (51). - Wolfdieter Lang, Dec 13 2018

More terms from Rémy Sigrist, Oct 16 2018

A322409 Compound tribonacci sequence with a(n) = A278040(A278040(n)), for n >= 0.

Original entry on oeis.org

5, 18, 29, 42, 49, 62, 73, 86, 99, 110, 123, 130, 143, 154, 167, 178, 191, 198, 211, 222, 235, 248, 259, 272, 279, 292, 303, 316, 323, 336, 347, 360, 373, 384, 397, 404, 417, 428, 441, 452, 465, 472, 485, 496, 509, 522, 533, 546, 553, 566, 577, 590, 603, 614, 627, 634, 647, 658, 671, 682, 695

Offset: 0

Wolfdieter Lang, Jan 02 2019

(a(n+1)) = A319968(n)-1 = A003145(A003145(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, A322408.

Cf. A003144, A003145, A003146.

a(n) = A(A(n)) = A(A(n) + 1) - 3 = 2*(A(n) + B(n)) + n + 3, for n >= 0, where A = A278040 and B = A278039. For a proof see the W. Lang link in A278040, Proposition 9, eq. (48).

A322410 Compound tribonacci sequence with a(n) = A278040(A278039(n)), for n >= 0.

Original entry on oeis.org

1, 8, 14, 21, 25, 32, 38, 45, 52, 58, 65, 69, 76, 82, 89, 95, 102, 106, 113, 119, 126, 133, 139, 146, 150, 157, 163, 170, 174, 181, 187, 194, 201, 207, 214, 218, 225, 231, 238, 244, 251, 255, 262, 268, 275, 282, 288, 295, 299, 306, 312, 319, 326, 332, 339, 343, 350, 356, 363, 369, 376

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

Cf. A278039, A278040, A322409.

Cf. A003144, A003145, A003146.

A(B(n)) = A(B(n) + 1) - 4 = A(n) + B(n) + n, for n >= 0, with A = A278040 and B = A278039. For a proof see the W. Lang link in A278040, Proposition 9, eq. (49).

a(n+1) = A319967(n)-1 = A003145(A003144(n))-1, the corresponding classical compound tribonacci sequence. - Michel Dekking, Apr 04 2019

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

Original entry on oeis.org

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).

cp's OEIS Frontend

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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A278041 The tribonacci representation of a(n) is obtained by appending 0,1,1 to the tribonacci representation of n (cf. A278038).

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

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A319968 a(n) = A003145(A003145(n)).

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A319967 a(n) = A003145(A003144(n)) where A003144 and A003145 are positions of '1' and '2' in the tribonacci word A092782.

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A322409 Compound tribonacci sequence with a(n) = A278040(A278040(n)), for n >= 0.

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A322410 Compound tribonacci sequence with a(n) = A278040(A278039(n)), for n >= 0.

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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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