A225330 -id:A225330 - cp's OEIS frontend

A225224 A continuous "look-and-say" sequence (without repetition, seed 1,1,1).

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

Offset: 1

Jean-Christophe Hervé, May 02 2013

A variant of the Conway's 'look-and-say' sequence A005150, without run cut-off. It describes at each step the preceding numbers taken altogether.
The sequence is better described as starting with three 1's: 1, 1, 1, and then 3, 1, and 1, 3, etc., as seed one creates a singular case: 1, then 1, 1, which can be continued either as 2, 1 (ignoring the aforesaid first 1, cf. A221646), or as 3, 1, considering twice the first one.
Contrary to the original look-and-say, this sequence is not base dependent, because figures or group of figures are not aggregated and read as numbers.
The sequence is determined by pairs. Terms of even ranks are counts while odd ranks are numbers.
As in the original look-and-say sequence, a(n) is always equal to 1, 2 or 3. The subsequence 3,3,3 never appears.
Two successive odd ranks cannot be equal, which implies that sequences of length three always begin on even rank and that two such sequences never follow each other.
Applying the look-and-say principle to the sequence itself, it is simply shift three ranks to the left.
With seed 2 (resp. 3), the sequence is A088203 (resp. A088204). These two sequences are shifted one rank left by the look-and-say transform.
With seed 2, the sequence A088203 is the concatenation of A006751 (original look-and-say method by blocks): this is because all blocks begin with 1 or 3 and end with 2 and therefore, there is no possible interaction between blocks after concatenation.

			The sequence starts with: 1, 1, 1
The first group has three 1's: 3, 1
The next group has one 3: 1, 3
The next group has two 1's: 2, 1
The next group has one 3: 1, 3
The next group has one 2: 1, 2
The next group has two 1's: 2, 1, etc.

J.-C. Hervé, Table of n, a(n) for n = 1..10000

Cf. A005150 (original look-and-say sequence).
Cf. A221646 (a close variant with seed 1).
Cf. A225212 (a variant with nested repetitions).
Cf. A088203 (seed 2), A088204 (seed 3).
Cf. A225330 (look-and-repeat).

/* computes first n terms in array a[] */
int *swys(int n) {
int a[n] ;
int see, say, c ;
a[0] = 1;
see = say = 0 ;
while( say < n-1 ) {
  c = 0 ;     /* count */
  dg = a[see] /* digit */
  if (say > 0) { /* not the first time */
    while (see <= say) {
      if (a[see]== dg)  c += 1 ;
      else break ;
      see += 1 ;
      }
    }
  else {
   c = 1 ;
    }
  a[++say] = c ;
  if (say < n-1) a[++say] = dg ;
  }
return(a);
}

n = 100; a[0] = 1; see = say = 0; While[ say < n - 1, c = 0; dg = a[see]; If[say > 0, While[ see <= say, If[a[see] == dg, c += 1, Break[]]; see += 1], c = 1]; a[++say] = c; If[say < n - 1, a[++say] = dg]]; Array[a, n, 0] (* Jean-François Alcover, Jul 11 2013, translated and adapted from J.-C. Hervé's C program *)

A225329 Look-and-repeat: similar to look-and-say except frequency is repeated.

Original entry on oeis.org

1, 111, 331, 223111, 222113331, 332221333111, 223332111333331, 222333112331553111, 332333221112223111225113331, 223112333222331332113331222115221333111

Offset: 1

Jean-Christophe Hervé, May 12 2013

Repeated frequency followed by digit-indication. Repeating the frequency allows 5 to appear, in addition to 1, 2 and 3 which are already contained in Conway's original look-and-say sequence. However, 4 still does not appear.
The sequence is determined by triples of digits. The first two terms of a triple are the repeated figure and the last term is the digit.
Therefore, sequences of form xy (x != y), xxyy can never appear. A fortiori, the sequence never contains series of four identical digits, but contains series of five 3, which make appear the 5's (55 and 5). However five 5's never appear. Proof: suppose it appears for the first time in a(n)-a(n+4); because of 'five five 5' in 55555, it would imply that 55555 appears form a smaller n, which is a contradiction. By the same argument, 555 also never appear.
Also 22222 or 11111 are impossible : 22222 would imply a preceding 22yy and 11111 a preceding 1x (x != 1), but both cannot exist.
All terms end with 1 (the seed) and, except the first two, begin with 2 or 3.

			The term after 331 is obtained by saying (repeating) two two 3, one one 1, which gives 223111.

Cf. A005150 (original look-and-say), A225224, A221646, A225212 (continuous look-and-say versions), A225330, A225331 (continuous look-and-repeat).

A225331 A continuous "look-and-repeat" sequence (method 2).

Original entry on oeis.org

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

Offset: 1

Jean-Christophe Hervé, May 12 2013

A variant of the 'look-and-repeat' sequence A225329, without run cut-off. It describes at each step the preceding digits by repeating the frequency number.
The sequence is determined by triples of digits. The first two terms of a triple are the repeated frequency and the last term is the digit.
There are different optional rules to build such a sequence. This method 2 never considers twice the already said digits.
With this rule and seed, a(n) is always equal to 1, 2, 3 or 5, and the sequence is the simple concatenation of the look-and-repeat sequence by block A225329. This is because all blocks of A225329 begin with 2 or 3 and end with 2 and therefore, there is no possible interaction between blocks after concatenation.
It never contains runs of exactly four identical digits (except the first four ones), but it does contain runs of five identical digits. However, five 5's never appear. Proof: suppose '55555' appears for the first time in a(n)..a(n+4); because of 'five five 5' in 55555, it would imply that 55555 appears from a smaller n, which is a contradiction.

			a(1) = 1, then a(2) = a(3) = a(4) = 1 (one one 1). Leaving out the first 1 already said, we now have three 1's, then a(5) = a(6) = 3, and a(7) = 1, etc.

Cf. A225330 (a close variant with 4's), A225329 (look-and-repeat by block), A005150 (original look-and-say), A225224, A221646, A225212 (continuous look-and-say versions).

A225332 A continuous "look-and-repeat" sequence (seed 2).

Original entry on oeis.org

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

Offset: 1

Jean-Christophe Hervé, May 12 2013

The 'look-and-repeat' sequence A225330, with seed 2. The variant A225331 with the same seed 2 gives this same sequence.
It describes at each step the preceding digits by repeating the frequency number.
The sequence is determined by triples of digits. The first two terms of a triple are the repeated count and the last term is the digit.
a(n) is always equal to 1, 2, 3, or 5. No series of four identical digits happens in the sequence, nor any of five 5's.
Applying the look-and-repeat principle to the sequence itself, it is simply shift one rank to the left.

			a(1) = 2, you then see "one 2" and repeating "one", a(2) = a(3) = 1 (one) and a(4) = 2; you have then two 1's, so 2, 2, 1; then three 2, so 3, 3, 1, etc.

Cf. A225330, A225331 (two variants with seed 1), A225329 (look-and-repeat by block), A005150 (original look-and-say), A225224, A221646, A225212 (continuous look-and-say versions).

A225333 Look-and-repeat: similar to look-and-say except frequency is repeated (seed 2).

Original entry on oeis.org

2, 112, 221112, 222331112, 332223331112, 223332333331112, 222333112553331112, 332333221112225333331112, 223112333222331332115553331112, 222113221112333332223111223112221335333331112

Offset: 1

Jean-Christophe Hervé, May 12 2013

Look-and-repeat sequence A225329 with seed 2.
Contains 1, 2, 3 and 5, but not 4.
All terms end with 2 (the seed) and, starting at the fourth, with 3331112, which makes the 5 appear.
All terms except the second begin with 2 or 3; this is a direct consequence of the look-and-repeat rule.

			The term after 112 is obtained by saying (repeating) two two 1, one one 2, which gives 221112.

Cf. A225329 (seed one), A225332 (continuous look-and-repeat, seed 2), A225330, A225331 (continuous look-and-repeat, seed 1).

Cf. A005150 (original look-and-repeat), A225224, A221646, A225212 (continuous look-and-repeat versions).

A225334 Look-and-repeat: similar to look-and-say except frequency is repeated (seed 3).

Original entry on oeis.org

3, 113, 221113, 222331113, 332223331113, 223332333331113, 222333112553331113, 332333221112225333331113, 223112333222331332115553331113, 222113221112333332223111223112221335333331113

Offset: 1

Jean-Christophe Hervé, May 12 2013

Look-and-repeat sequence A225329 with seed 3.
Contains 1, 2, 3 and 5, but not 4.
All terms end with 3 (the seed) and, starting at the fourth, with 3331113, which makes the 5 appear.
All terms except the second begin with 2 or 3; it is a direct consequence of the look-and-repeat rule.
Sequence A225333, the look-and-repeat sequence with seed 2 is almost the same, differing only in the last digit of each term (2 instead of 3 here). Indeed, any one-digit seed except 1 leads to essentially the same sequence with all terms identical except the last one = the seed.

			The term after 113 is obtained by saying (repeating) two two 1, one one 3, which gives 221113.

Cf. A225329 (seed one), A225333 (seed two), A225332 (continuous look-and-repeat, seed 2), A225330, A225331 (continuous look-and-repeat, seed 1).

Cf. A005150 (original look-and-say), A225224, A221646, A225212 (continuous look-and-say versions).

a(n) = A225333(n) + 1.

A225335 A continuous "look-and-repeat" sequence (seed 3).

Original entry on oeis.org

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

Offset: 1

Jean-Christophe Hervé, May 12 2013

The 'look-and-repeat' sequence A225330, with seed 3. The variant A225331 with the same seed 3 gives this same sequence.
It describes at each step the preceding digits by repeating the frequency number.
The sequence is determined by triples of digits. The first two terms of a triple are the repeated frequency and the last term is the digit.
a(n) is always equal to 1, 2, 3, or 5. No series of four identical digits happens in the sequence, nor any of five 5's.
Applying the look-and-repeat principle to the sequence itself, it is simply shift one rank to the left.

			a(1) = 3, you then see "one 3" and repeating "one", a(2) = a(3) = 1 (one) and a(4) = 3; you have then two 1's, so 2,2,1 ; then one 3, so 1,1,3, etc.

Cf. A225330, A225331 (two variants with seed 1), A225331 (seed 2), A225329, A225333, A225334 (look-and-repeat by block with resp. seed 1, seed 2, seed 3).

Cf. A005150 (original look-and-say), A225224, A221646, A225212 (continuous look-and-say versions).

cp's OEIS Frontend

A225224 A continuous "look-and-say" sequence (without repetition, seed 1,1,1).

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A225329 Look-and-repeat: similar to look-and-say except frequency is repeated.

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A225331 A continuous "look-and-repeat" sequence (method 2).

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A225332 A continuous "look-and-repeat" sequence (seed 2).

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A225333 Look-and-repeat: similar to look-and-say except frequency is repeated (seed 2).

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A225334 Look-and-repeat: similar to look-and-say except frequency is repeated (seed 3).

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A225335 A continuous "look-and-repeat" sequence (seed 3).

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