OFFSET
1,4
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.
LINKS
J.-C. Hervé, Table of n, a(n) for n = 1..10000
EXAMPLE
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.
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 *)
PROG
(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);
}
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jean-Christophe Hervé, May 02 2013
STATUS
approved