A306211 -id:A306211 - cp's OEIS frontend

A306215 a(n) gives the length of A306211 after n generations.

1, 2, 3, 5, 8, 13, 22, 37, 61, 100, 162, 260, 416, 663, 1053, 1664, 2617, 4102, 6416, 10029, 15684, 24561, 38531, 60560, 95334, 150238, 236878, 373449, 588441, 926449, 1457335, 2290726, 3599062, 5654352, 8886976, 13979961, 22020411, 34743031, 54922083

Offset: 1

Peter Kagey, Jan 29 2019

nonn

n | A306211 after n generations
--+-----------------------------
| [1]
| [1, 1]
| [1, 1, 2]
| [1, 1, 2, 2, 1]
| [1, 1, 2, 2, 1, 2, 2, 1]
| [1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1]
| [1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1]

Benjamin Chaffin, Table of n, a(n) for n = 1..65

Cf. A306211.

See A323475 for first differences.

a(27)-a(39) from Rémy Sigrist, Jan 29 2019

A306222 Positions of 4's in A306211.

Original entry on oeis.org

60, 84, 93, 96, 123, 132, 135, 147, 150, 156, 162, 185, 194, 197, 209, 212, 218, 224, 232, 235, 241, 247, 249, 283, 292, 295, 307, 310, 316, 322, 330, 333, 339, 345, 347, 366, 369, 375, 381, 383, 396, 439, 448, 451, 463, 466, 472, 478, 486, 489, 495, 501, 503

Offset: 1

Rémy Sigrist, Jan 30 2019

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..25000

Cf. A306211, A306223, A323476.

A306223 Positions of 5's in A306211.

Original entry on oeis.org

255, 353, 389, 402, 509, 545, 558, 603, 616, 638, 660, 662, 756, 792, 805, 850, 863, 885, 907, 909, 941, 954, 976, 998, 1000, 1009, 1031, 1033, 1043, 1045, 1049, 1146, 1182, 1195, 1240, 1253, 1275, 1297, 1299, 1331, 1344, 1366, 1388, 1390, 1399, 1421, 1423

Offset: 1

Rémy Sigrist, Jan 30 2019

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..25000

Cf. A306211, A306222, A323476.

A323475 Number of new terms added at n-th generation of A306211.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 15, 24, 39, 62, 98, 156, 247, 390, 611, 953, 1485, 2314, 3613, 5655, 8877, 13970, 22029, 34774, 54904, 86640, 136571, 214992, 338008, 530886, 833391, 1308336, 2055290, 3232624, 5092985, 8040450, 12722620, 20179052

Offset: 0

N. J. A. Sloane, Jan 29 2019

nonn

a(0)=1 followed by first differences of A306215.

Benjamin Chaffin, Table of n, a(n) for n = 0..64

Cf. A306211, A306215.

A323476 Positions of 3's in A306211.

Original entry on oeis.org

37, 52, 58, 76, 82, 91, 115, 121, 130, 145, 177, 183, 192, 207, 230, 257, 275, 281, 290, 305, 328, 355, 364, 391, 404, 410, 416, 431, 437, 446, 461, 484, 511, 520, 547, 560, 566, 572, 578, 605, 618, 624, 630, 640, 646, 654, 678, 684, 693, 708, 731, 758, 767

Offset: 1

N. J. A. Sloane, Jan 29 2019

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..25000
Rémy Sigrist, C program for A323476

Cf. A306211, A306222, A306223.

C
```
See Links section.
```

seq[n_] := seq[n] = If[n==1, {1}, Join[seq[n-1], Length /@ Split[seq[n-1]]] ];
Position[seq[26], 3] // Flatten (* Jean-François Alcover, Jul 19 2022 *)

A323477 Successive generations of A306211, in compressed notation.

Original entry on oeis.org

1, 11, 112, 11221, 11221221, 1122122122121, 1122122122121221212111, 1122122122121221212111221212111211113, 1122122122121221212111221212111211113221212111211113211113141

Offset: 1

N. J. A. Sloane, Jan 31 2019

"Compressed" means the separating commas have been omitted. This will only work as long as the terms of A306211 are at most 9. However, we know from Chaffin's work (see A306211) that this is true at least for the first 10228800161220 terms of A306211.

Cf. A306211, A306215 (lengths), A323478 (increments).

s[n_] := If[n == 1, {1}, s[n] = Join[s[n-1], Length /@ Split[s[n-1]]]];
a[n_] := FromDigits[s[n]];
Array[a, 9] (* Jean-François Alcover, Feb 24 2021 *)

A323478 a(1)=1; thereafter, a(n) = string appended to (n-1)-st generation of A306211 to get the n-th generation.

Original entry on oeis.org

1, 1, 2, 21, 221, 22121, 221212111, 221212111211113, 221212111211113211113141, 221212111211113211113141211113141141111, 22121211121111321111314121111314114111121111314114111114111214

Offset: 1

N. J. A. Sloane, Jan 31 2019

The concatenation of these terms is the sequence A306211 itself.
The concatenation A323477(n-1) followed by a(n) is A323477(n). For example, if n=6, 11221221.22121 = 1122122122121.
The strings a(n) are converging to A323826, the RUNS transform of A306211.

N. J. A. Sloane, Table of n, a(n) for n = 1..17

Cf. A306211, A323477, A306215, A323475 (lengths), A323826 (limiting string).

A323479 Irregular triangle read by rows: T(n,k) (n>=1) = number of occurrences of k in n-th generation of A306211.

Original entry on oeis.org

1, 2, 2, 1, 3, 2, 4, 4, 6, 7, 11, 11, 20, 16, 1, 35, 22, 3, 1, 61, 29, 6, 4, 103, 38, 10, 11, 170, 50, 16, 23, 1, 278, 66, 27, 41, 4, 451, 87, 46, 67, 12, 728, 114, 77, 103, 31, 1165, 149, 127, 152, 71, 1849, 194, 210, 216, 148, 2916, 251, 354, 297, 284, 4577

Offset: 1

N. J. A. Sloane, Jan 31 2019

			Triangle begins:
1,
2,
2, 1,
3, 2,
4, 4,
6, 7,
11, 11,
20, 16, 1,
35, 22, 3, 1,
61, 29, 6, 4,
103, 38, 10, 11,
170, 50, 16, 23, 1,
...

Lars Blomberg, Table of n, a(n) for n = 1..247 (The first 55 generations)

Cf. A306211, A306215 (row sums), A323477, A323481 (first column).

a(20)-a(63) from Lars Blomberg, Feb 13 2019

A323481 Number of 1's in n-th generation of A306211.

Original entry on oeis.org

1, 2, 2, 3, 4, 6, 11, 20, 35, 61, 103, 170, 278, 451, 728, 1165, 1849, 2916, 4577, 7165, 11205, 17531, 27467, 43119, 67823, 106854, 168523, 265878, 419356, 660919, 1040546, 1636458, 2571310, 4037926, 6340362, 9959607, 15658964, 24653360, 38881723, 61445514

Offset: 1

N. J. A. Sloane, Jan 31 2019

nonn

Benjamin Chaffin, Table of n, a(n) for n = 1..65

Cf. A306211.

First column of A323479.

More terms from Benjamin Chaffin, Feb 07 2019

A306333 Last term in each generation of A306211.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 4, 1, 3, 1, 1, 4, 1, 3, 1, 3, 1, 3, 1, 1, 4, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 4, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 4, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3

Offset: 1

Benjamin Chaffin, Feb 08 2019

nonn

The first 65 generations of A306211 were computed explicitly. Since the tail of each generation is the RUNS of the tail of the previous generation, if we start with a complete generation then we can get just the tail of some further generations by iterating RUNS on it. This allowed the computation of the remaining 55 terms.

Generations 11, 16, 25, 42, and 75 end in a 4. The number of generations in between those which end in a 4 is thus 4, 8, 16, 32.

Benjamin Chaffin, Table of n, a(n) for n = 1..120

Cf. A306211.

cp's OEIS Frontend

A306215 a(n) gives the length of A306211 after n generations.

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A306222 Positions of 4's in A306211.

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A306223 Positions of 5's in A306211.

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A323475 Number of new terms added at n-th generation of A306211.

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A323476 Positions of 3's in A306211.

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C

Mathematica

A323477 Successive generations of A306211, in compressed notation.

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Programs

Mathematica

A323478 a(1)=1; thereafter, a(n) = string appended to (n-1)-st generation of A306211 to get the n-th generation.

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A323479 Irregular triangle read by rows: T(n,k) (n>=1) = number of occurrences of k in n-th generation of A306211.

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Examples

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A323481 Number of 1's in n-th generation of A306211.

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Extensions

A306333 Last term in each generation of A306211.

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