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

Author

Jean-Christophe Hervé, May 02 2013

Keywords

• nonn
• easy

Comments

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.

Examples

			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.
		

Links

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

Crossrefs

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).

Programs

C

/* 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);
}

Mathematica

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 *)