A306211 -id:A306211 - cp's OEIS frontend

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.

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

N. J. A. Sloane, Jan 31 2019

			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,
...

Lars Blomberg, Table of n, a(n) for n = 1..251 (The first 55 generations)

Cf. A306211, A306215, A323477.

a(23)-a(68) from Lars Blomberg, Feb 13 2019

A323826 RUNS transform of A306211.

Original entry on oeis.org

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, 1, 4, 1, 1, 4, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 4, 1, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 4, 2, 1, 1, 1, 1, 3, 1, 4, 1, 1, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 4, 1, 4, 1, 1, 1, 2, 1, 5, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 3

Offset: 1

N. J. A. Sloane, Feb 01 2019

nonn

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003.

Cf. A306211.

A323827 Apply Lenormand's "raboter" operation to A306211.

Original entry on oeis.org

1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Feb 01 2019

nonn

The operation shortens each run of consecutive equal terms by one term (runs of length 1 vanish).

Rémy Sigrist, Table of n, a(n) for n = 1..16502
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003.
Rémy Sigrist, PARI program for A323827

Cf. A306211, A318921.

PARI
```
See Links section.
```

More terms from Rémy Sigrist, Nov 11 2020

A323829 Position where n first appears in A306211, or -1 if n never appears.

Original entry on oeis.org

1, 3, 37, 60, 255

Offset: 1

N. J. A. Sloane, Feb 01 2019

If 6 does appear in A306211, this happens after term 10228800161220 (see comments in A306211).

Cf. A306211.

A381587 a(1) = 1; thereafter the sequence is extended by iteratively appending the run length transform of the reverse of the sequence thus far.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3

Offset: 1

Neal Gersh Tolunsky, Feb 27 2025

The run length transform replaces each run of consecutive equal values with a single value representing the length of that run.
Is 2 the greatest even number in the sequence?
It appears that the limit of the rows read in reverse equals A306346 (ignoring the initial terms). - Paul D. Hanna, Mar 03 2025

			Irregular triangle begins:
  1;
  1;
  2;
  1,2;
  1,1,1,2;
  1,3,1,1,1,2;
  1,3,1,1,1,3,1,1,1,2;
  1,3,1,3,1,1,1,3,1,1,1,3,1,1,1,2;
  1,3,1,3,1,3,1,1,1,1,1,3,1,3,1,1,1,3,1,1,1,3,1,1,1,2;
  ...

Paul D. Hanna, Table of n, a(n) for n = 1..9259 (first 20 rows)

Cf. A381357 (row lengths), A381358 (row sums), A381356 (limit of rows), A306346.

Cf. A306211.

\\ From Paul D. Hanna, Mar 03 2025: (Start)
\\ RUNS(V) Returns vector of run lengths in vector V:
{RUNS(V) = my(R=[],c=1);if(#V>1, for(n=2,#V, if(V[n]==V[n-1], c=c+1, R=concat(R,c); c=1))); R=concat(R,c)}
\\ REV(V) Reverses order of vector V:
{REV(V) = Vec(Polrev(Ser(V)))}
\\ Generates N rows as a vector A of row vectors
{N=15; A=vector(N);A[1]=[1];A[2]=[1];A[3]=[2];
for(n=3,#A-1, A[n+1] = concat(RUNS(REV(A[n])),A[n]);); A}
for(n=1,N,print(A[n])) \\ Print N rows of this triangle (End)

A365836 a(1)=1; thereafter extend the sequence by appending its nonincreasing run transform, recompute the nonincreasing run transform, append it, and so on.

Original entry on oeis.org

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

Offset: 1

Neal Gersh Tolunsky, Sep 19 2023

The nonincreasing run transform replaces each run of weakly decreasing terms with a single value which is the length of that run.

			Irregular triangle in which each row (after the initial 1) is the nonincreasing run transform of the concatenation of the previous rows:
  1;
  1;
  2;
  2, 1;
  2, 3;
  2, 3, 1, 1;
  2, 3, 1, 2, 3;
  2, 3, 1, 2, 3, 1, 2, 1, 1;
  2, 3, 1, 2, 3, 1, 2, 1, 2, 2, 1, 2, 3;
  2, 3, 1, 2, 3, 1, 2, 1, 2, 2, 1, 2, 3, 1, 2, 1, 2, 2, 3, 1, 1;
  ...

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10632 (first 25 rows)
Neal Gersh Tolunsky, Ordinal transform on the first 35 iterations of the sequence

Cf. A365838, A306211.

A365838 a(1)=2; thereafter extend the sequence by appending its nondecreasing run transform, recompute the nondecreasing run transform, append it, and so on.

Original entry on oeis.org

2, 1, 1, 1, 1, 3, 1, 5, 1, 5, 2, 1, 5, 2, 2, 1, 1, 5, 2, 2, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 2, 3, 2, 3, 1, 1, 5, 2, 2, 1, 2, 2, 3, 2, 3, 3, 2, 4, 2, 1, 1, 5, 2, 2, 1, 2, 2, 3, 2, 3, 3, 2, 4, 2, 3, 2, 4, 3, 2, 1, 1, 1, 5, 2, 2, 1, 2, 2, 3, 2, 3, 3, 2, 4, 2, 3, 2, 4, 3, 2, 1, 3, 2, 4, 3, 2, 2, 2, 1, 1

Offset: 1

Neal Gersh Tolunsky, Sep 20 2023

The nondecreasing run transform replaces each run of weakly increasing terms with a single value which is the length of that run.

			Irregular triangle in which each row (after the initial 2) is the nondecreasing run transform of the concatenation of the previous rows:
  2;
  1;
  1, 1;
  1, 3;
  1, 5;
  1, 5, 2;
  1, 5, 2, 2, 1;
  1, 5, 2, 2, 1, 2, 2, 1;
  1, 5, 2, 2, 1, 2, 2, 3, 2, 3, 1;
  1, 5, 2, 2, 1, 2, 2, 3, 2, 3, 3, 2, 4, 2, 1;
  ...

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..9294 (first 25 rows)
Neal Gersh Tolunsky, Ordinal transform on the first 35 iterations of the sequence

Cf. A365836, A306211.

A306216 Successive concatenation of the current sequence with the first differences of the sequence, a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 1, -1, 1, -1, 0, -1, 0, -1, 1, -1, 1, -1, 1, -1, 1, -1, 2, -2, 2, -2, 0, -1, 0, -1, 1, -1, 1, -1, 1, -1, 1, -1, 2, -2, 2, -2, 1, -1, 1, -1, 2, -2, 2, -2, 2, -2, 2, -2, 3, -4, 4, -4, 0, -1, 0, -1, 1, -1, 1, -1, 1, -1

Offset: 1

Peter Kagey, Jan 29 2019

n | generation             | first differences
--+------------------------+-------------------
1 | [1,1]                  | [0]
2 | [1,1,0]                | [0,-1]
3 | [1,1,0,0,-1]           | [0,-1,0,-1]
4 | [1,1,0,0,-1,0,-1,0,-1] | [0,-1,0,-1,1,-1,1,-1]

Peter Kagey, Table of n, a(n) for n = 1..8193 (first 14 generations)

Cf. A107946, A306211.

a306216_list = 1 : 1 : concat (Data.List.unfoldr nextGeneration [1,1]) where
  nextGeneration l = Just (diff l, l ++ diff l)
  diff xs =  zipWith subtract xs (tail xs)

Nest[Join[#, Differences@ #] &, {1, 1}, 7] (* Michael De Vlieger, Jan 29 2019 *)

generations = 10
(1...generations).reduce([1,1]) do |s, _|
  s += s.each_cons(2).map { |a, b| b - a }
end

cp's OEIS Frontend

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.

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Extensions

A323826 RUNS transform of A306211.

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A323827 Apply Lenormand's "raboter" operation to A306211.

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PARI

Extensions

A323829 Position where n first appears in A306211, or -1 if n never appears.

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A381587 a(1) = 1; thereafter the sequence is extended by iteratively appending the run length transform of the reverse of the sequence thus far.

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

PARI

A365836 a(1)=1; thereafter extend the sequence by appending its nonincreasing run transform, recompute the nonincreasing run transform, append it, and so on.

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Keywords

Comments

Examples

Links

Crossrefs

A365838 a(1)=2; thereafter extend the sequence by appending its nondecreasing run transform, recompute the nondecreasing run transform, append it, and so on.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A306216 Successive concatenation of the current sequence with the first differences of the sequence, a(1) = a(2) = 1.

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Author

Keywords

Comments

Links

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Programs

Haskell

Mathematica

Ruby