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

%I A225224 #55 Apr 11 2018 02:59:40
%S A225224 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,
%T A225224 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,
%U A225224 3,2,2,2,1,2,3,2,1,1,2,1,1,1,3,1,2,1,1
%N A225224 A continuous "look-and-say" sequence (without repetition, seed 1,1,1).
%C A225224 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.
%C A225224 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.
%C A225224 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.
%C A225224 The sequence is determined by pairs. Terms of even ranks are counts while odd ranks are numbers.
%C A225224 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.
%C A225224 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.
%C A225224 Applying the look-and-say principle to the sequence itself, it is simply shift three ranks to the left.
%C A225224 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.
%C A225224 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.
%H A225224 J.-C. Hervé, <a href="/A225224/b225224.txt">Table of n, a(n) for n = 1..10000</a>
%e A225224 The sequence starts with: 1, 1, 1
%e A225224 The first group has three 1's: 3, 1
%e A225224 The next group has one 3: 1, 3
%e A225224 The next group has two 1's: 2, 1
%e A225224 The next group has one 3: 1, 3
%e A225224 The next group has one 2: 1, 2
%e A225224 The next group has two 1's: 2, 1, etc.
%t A225224 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 *)
%o A225224 (C) /* computes first n terms in array a[] */
%o A225224 int *swys(int n) {
%o A225224 int a[n] ;
%o A225224 int see, say, c ;
%o A225224 a[0] = 1;
%o A225224 see = say = 0 ;
%o A225224 while( say < n-1 ) {
%o A225224   c = 0 ;     /* count */
%o A225224   dg = a[see] /* digit */
%o A225224   if (say > 0) { /* not the first time */
%o A225224     while (see <= say) {
%o A225224       if (a[see]== dg)  c += 1 ;
%o A225224       else break ;
%o A225224       see += 1 ;
%o A225224       }
%o A225224     }
%o A225224   else {
%o A225224    c = 1 ;
%o A225224     }
%o A225224   a[++say] = c ;
%o A225224   if (say < n-1) a[++say] = dg ;
%o A225224   }
%o A225224 return(a);
%o A225224 }
%Y A225224 Cf. A005150 (original look-and-say sequence).
%Y A225224 Cf. A221646 (a close variant with seed 1).
%Y A225224 Cf. A225212 (a variant with nested repetitions).
%Y A225224 Cf. A088203 (seed 2), A088204 (seed 3).
%Y A225224 Cf. A225330 (look-and-repeat).
%K A225224 nonn,easy
%O A225224 1,4
%A A225224 _Jean-Christophe Hervé_, May 02 2013