cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Wolfdieter Lang, Jan 02 2019

Keywords

Comments

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

Crossrefs

Cf. A278039, A278040, A322409.
Cf. A003144, A003145, A003146.

Formula

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

Views

Author

Wolfdieter Lang, Jan 02 2019

Keywords

Comments

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

Links

Crossrefs

Cf. A278040, A278041, A322410.

Formula

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

Views

Author

Wolfdieter Lang, Jan 02 2019

Keywords

Comments

(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

Crossrefs

Cf. A278040, A278041, A322411.

Formula

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

Views

Author

Wolfdieter Lang, Jan 02 2019

Keywords

Comments

(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

Crossrefs

Cf. A278039, A278040, A278041, A322412.
Cf. A003144, A003145, A003146.

Formula

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

Views

Author

Wolfdieter Lang, Jan 02 2019

Keywords

Comments

(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

Crossrefs

Cf. A278039, A278040, A278041, A322413.
Cf. A003144, A003145, A003146.

Formula

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).

A245553 A Rauzy fractal sequence: trajectory of 1 under morphism 1 -> 2,3; 2 -> 3; 3 -> 1.

Original entry on oeis.org

1, 2, 3, 2, 3, 3, 1, 2, 3, 3, 1, 3, 1, 1, 2, 3, 2, 3, 3, 1, 3, 1, 1, 2, 3, 3, 1, 1, 2, 3, 1, 2, 3, 2, 3, 3, 1, 2, 3, 3, 1, 3, 1, 1, 2, 3, 3, 1, 1, 2, 3, 1, 2, 3, 2, 3, 3, 1, 3, 1, 1, 2, 3, 1, 2, 3, 2, 3, 3, 1, 1, 2, 3, 2, 3, 3, 1, 2, 3, 3, 1, 3, 1, 1, 2, 3
Offset: 1

Views

Author

N. J. A. Sloane, Aug 03 2014

Keywords

Links

Crossrefs

Cf. A092782, A245554, A105083.

Programs

  • Mathematica
    Nest[ Function[ l, {Flatten[(l /. {1 -> {2, 3}, 2 -> {3}, 3 -> {1}})] }], {1}, 15]

A245554 A Rauzy fractal sequence: trajectory of 1 under morphism 1 -> 1,2,1,3; 2 -> 3; 3 -> 1.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 2, 1, 3, 1, 1, 1, 2, 1, 3, 3, 1, 2, 1, 3, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 3, 1, 2, 1, 3, 1, 1, 1, 2, 1, 3, 3, 1, 2, 1, 3, 1, 1, 2, 1, 3, 1, 2, 1, 3, 3, 1, 2, 1, 3, 1, 1, 2, 1, 3, 3, 1, 2, 1, 3, 1, 1, 2, 1, 3, 3, 1, 2, 1, 3, 1, 1, 1, 2, 1, 3, 3, 1, 2
Offset: 1

Views

Author

N. J. A. Sloane, Aug 03 2014

Keywords

Links

Crossrefs

Cf. A092782, A245553, A105083.

Programs

  • Mathematica
    Nest[ Function[ l, {Flatten[(l /. {1 -> {1, 2, 1, 3}, 2 -> {3}, 3 -> {1}})] }], {1}, 9]

A277735 Unique fixed point of the morphism 0 -> 01, 1 -> 20, 2 -> 0.

Original entry on oeis.org

0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1
Offset: 1

Views

Author

N. J. A. Sloane, Nov 07 2016

Keywords

Comments

From Clark Kimberling, May 21 2017: (Start)
Let u be the sequence of positions of 0, and likewise, v for 1 and w for 2. Let U, V, W be the limits of u(n)/n, v(n)/n, w(n)/n, respectively. Then 1/U + 1/V + 1/W = 1, where
U = 1.8392867552141611325518525646532866...,
V = U^2 = 3.3829757679062374941227085364...,
W = U^3 = 6.2222625231203986266745611011....
If n >=2, then u(n) - u(n-1) is in {1,2,3}, v(n) - v(n-1) is in {2,4,5}, and w(n) - w(n-1) is in {4,7,9}. (u = A277736, v = A277737, w = A277738). (End)
Although I believe the assertions in Kimberling's comment above to be correct, these results are quite tricky to prove, and unless a formal proof is supplied at present these assertions must be regarded as conjectures. - N. J. A. Sloane, Aug 20 2018
From Michel Dekking, Oct 03 2019: (Start)
Here is a proof of Clark Kimberling's conjectures (and more).
The incidence matrix of the defining morphism
sigma: 0 -> 01, 1 -> 20, 2 -> 0
is the same as the incidence matrix of the tribonacci morphism
0 -> 01, 1 -> 02, 2 -> 0
(see A080843 and/or A092782).
This implies that the frequencies f0, f1 and f2 of the letters 0,1, and 2 in (a(n)) are the same as the corresponding frequencies in the tribonacci word, which are 1/t, 1/t^2 and 1/t^3 (see, e.g., A092782).
Since U = 1/f0, V = 1/f1, and W = 1/f2, we conclude that
U = t = A058265, V = t^2 = A276800 and W = t^3 = A276801.
The statements on the difference sequences u, v, and w of the positions of 0,1, and 2 are easily verified by applying sigma to the return words of these three letters.
Here the return words of an arbitrary word w in a sequence x are all the words occurring in x with prefix w that do not have other occurrences of w in them.
The return words of 0 are 0, 01, and 012, which indeed have length 1, 2
and 3. Since
sigma(0) = 01, sigma(1) = 0120, and sigma(012) = 01200,
one sees that u is the unique fixed point of the morphism
1 -> 2, 2-> 31, 3 ->311.
With a little more work, passing to sigma^2, and rotating, one can show that v is the unique fixed point of the morphism
2->52, 4->5224, 5->52244 .
Similarly, w is the unique fixed point of the morphism
4->94, 7->9447, 9->94477.
Interestingly, the three morphisms having u,v, and w as fixed point are essentially the same morphism (were we replaced sigma by sigma^2) with standard form
1->12, 2->1223, 3->12233.
(End)
The kind of phenomenon observed at the end of the previous comment holds in a very strong way for the tribonacci word. See Theorem 5.1. in the paper by Huang and Wen. - Michel Dekking, Oct 04 2019

Links

Crossrefs

Cf. A277736, A277737, A277738.
Equals A100619(n)-1.

Programs

  • Maple
    with(ListTools);
T:=proc(S) Flatten(subs( {0=[0,1], 1=[2,0], 2=[0]}, S)); end;
S:=[0];
for n from 1 to 10 do S:=T(S); od:
S;
  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {2, 0}, 2 -> 0}] &, {0}, 10] (* A277735 *)
Flatten[Position[s, 0]] (* A277736 *)
Flatten[Position[s, 1]] (* A277737 *)
Flatten[Position[s, 2]] (* A277738 *)
(* Clark Kimberling, May 21 2017 *)

Extensions

Name clarified by Michel Dekking, Oct 03 2019

A316324 Indices of 1's in A305389.

Original entry on oeis.org

0, 5, 9, 14, 19, 23, 28, 31, 36, 40, 45, 49, 54, 57, 62, 66, 71, 76, 80, 85, 88, 93, 97, 102, 107, 111, 116, 119, 124, 128, 133, 137, 142, 145, 150, 154, 159, 164, 168, 173, 176, 181, 185, 190, 193, 198, 202, 207, 212, 216, 221, 224, 229, 233, 238, 242, 247, 250, 255, 259, 264, 269, 273, 278
Offset: 1

Views

Author

N. J. A. Sloane, Jul 09 2018

Keywords

Links

Crossrefs

Cf. A305389.
A316324, A316325, A316326 have the same relation to A305389 as A003144, A003145, A003146 do to the ternary tribonacci word A080843 (or A092782).

Programs

  • Maple
    f:= 'f':
f(1):= 2: f(2):= 3: f(3):= (1,2,3):
S:= [1];
for i from 1 to 5 do T:= map(f,S); U:= map(f,T); S:= map(f,U); od:
select(t -> S[t+1]=1, [$0..nops(S)-1]); # Robert Israel, May 07 2019

A316325 Indices of 2's in A305389.

Original entry on oeis.org

1, 3, 6, 10, 12, 15, 17, 20, 24, 26, 29, 32, 34, 37, 41, 43, 46, 50, 52, 55, 58, 60, 63, 67, 69, 72, 74, 77, 81, 83, 86, 89, 91, 94, 98, 100, 103, 105, 108, 112, 114, 117, 120, 122, 125, 129, 131, 134, 138, 140, 143, 146, 148, 151, 155, 157, 160, 162, 165
Offset: 1

Views

Author

N. J. A. Sloane, Jul 09 2018

Keywords

Links

Crossrefs

Cf. A305389.
A316324, A316325, A316326 have the same relation to A305389 as A003144, A003145, A003146 do to the ternary tribonacci word A080843 (or A092782).

Programs

  • Maple
    f:= 'f':
f(1):= 2: f(2):= 3: f(3):= (1,2,3):
S:= [1];
for i from 1 to 5 do T:= map(f,S); U:= map(f,T); S:= map(f,U); od:
select(t -> S[t+1]=2, [$0..nops(S)-1]); # Robert Israel, May 07 2019
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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