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.

Showing 1-4 of 4 results.

A306211 Concatenation of the current sequence with the lengths of the runs in the sequence, with a(1) = 1.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 1, 4, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 1, 4, 1, 2, 1, 1, 1, 1, 3
Offset: 1

Views

Author

A. D. Skovgaard, Jan 29 2019

Keywords

Comments

Conjecture: All terms are less than or equal to 5. - Peter Kagey, Jan 29 2019
Conjecture: Every number appears! (Based on the analogy with the somewhat similar sequence A090822, where the first 5 appeared at around 10^(10^23) steps). - N. J. A. Sloane, Jan 29 2019
An alternative definition: Start with 1, extend the sequence by appending its RUNS transform, recompute the RUNS transform, append it, repeat. - N. J. A. Sloane, Jan 29 2019
The first time we see 1, 2, 3, 4, 5 is at n=1, 3, 37, 60, 255 (A323829). After 65 generations (10228800161220 terms) the largest term is 5. The relative frequencies of 1..5 are roughly 0.71, 6.7e-9, 0.23, 1.6e-8, 0.061. 2s and 4s appear to get rarer as n increases. - Benjamin Chaffin, Feb 07 2019
If we record the successive RUNS transforms and concatenate them, we get 1; 2; 2, 1; 2, 2, 1; 2, 2, 1, 2, 1; ..., which is this sequence without the initial 1. - A. D. Skovgaard, Jan 30 2019 (Rephrased by N. J. A. Sloane, Jan 30 2019)

Examples

			a(2) = 1, since there is a run of length 1 at a(1).
a(3) = 2, since there is a run of length 2 at a(1..2).
a(4..5) = 2, 1, since the runs are as follows:
  1, 1, 2  a(1..3)
  \__/  |
  2,    1  a(4..5)
a(37) = 3, since a(20..22) = 1, 1, 1.
Steps in construction:
  [1]  initial sequence
  [1]  its run length
 .
  [1, 1] concatenation of above is new sequence
  [2]  its run length
 .
  [1, 1, 2] concatenation of above is new sequence
  [2, 1]  its run lengths
 .
  [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]
 .
  [1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1]
  [2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3]
 .
  [1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3]
From _N. J. A. Sloane_, Jan 31 2019: (Start)
The first 9 generations, in compressed notation (see A323477) are:
  1
  11
  112
  11221
  11221221
  1122122122121
  1122122122121221212111
  1122122122121221212111221212111211113
  1122122122121221212111221212111211113221212111211113211113141
  ... (End)

Links

Crossrefs

Cf. A000002, A107946, A306215, A090822.
Positions of 3's, 4's, 5's: A323476, A306222, A306223.
Successive generations: A323477, A323478, A306215, A323475, A306333.
See also A323479, A323480, A323481, A323826 (RUNS transform), A323827, A323829 (where n first appears).

Programs

  • Haskell
    group [] = []
group (x:xs)= (x:ys):group zs where (ys,zs) = span (==x) xs
a306211_next_gen xs = xs ++ (map length $ group xs)
a306211_gen 1 = [1]
a306211_gen n = a306211_next_gen $ a306211_gen (n-1)
a306211 n = a306211_gen n !! (n-1)
-- Jean-François Antoniotti, Jan 31 2021
  • Mathematica
    seq[n_] := seq[n] = If[n==1, {1}, Join[seq[n-1], Length /@ Split[seq[n-1]]]];
seq[10] (* Jean-François Alcover, Jul 19 2022 *)

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

Views

Author

N. J. A. Sloane, Jan 31 2019

Keywords

Comments

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.

Links

Crossrefs

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

Views

Author

N. J. A. Sloane, Jan 31 2019

Keywords

Examples

			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,
...

Links

Crossrefs

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

Extensions

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

A323480 Irregular triangle read by rows: T(n,k) (n>=1) = number of runs of length k (k>=1) in n-th generation of A306211.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 2, 3, 5, 4, 9, 5, 1, 15, 6, 2, 1, 26, 7, 3, 3, 42, 9, 4, 7, 67, 12, 6, 12, 1, 108, 16, 11, 18, 3, 173, 21, 19, 26, 8, 277, 27, 31, 36, 19, 437, 35, 50, 49, 40, 684, 45, 83, 64, 77, 1067, 57, 144, 81, 136, 1661, 71, 255, 101, 226, 2588, 87
Offset: 1

Views

Author

N. J. A. Sloane, Jan 31 2019

Keywords

Examples

			Triangle begins:
1,
0, 1,
1, 1,
1, 2,
2, 3,
5, 4,
9, 5, 1,
15, 6, 2, 1,
26, 7, 3, 3,
42, 9, 4, 7,
67, 12, 6, 12, 1,
108, 16, 11, 18, 3,
...

Links

Crossrefs

Cf. A306211, A306215, A323477.

Extensions

a(23)-a(68) from Lars Blomberg, Feb 13 2019
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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