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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.

a(n) is always equal to 1, 2, 3, 4 or 5.

However, the occurrence of 4 is specific to this variant (method and seed), and only due to the initial sequence of four 1's. No other series of four identical digit happens in the sequence.

There are different optional rules to build such a sequence. This method 1 does not consider already said digits, unless if the length of the sequence of repeated figures to which they belongs change : this happens only once at the beginning, with the first 1 which is considered twice (and this brings up the 4): 1 -> 1,1,1 -> 4,4,1. The variant A225330 never considers already said digit (and does not contain 4). With other seeds (for example, 2 or 3), this special case at the beginning does not arise, and both variants coincide (and do not contain 4).

Applying the look-and-repeat principle to the sequence itself, it is simply shift four ranks to the left.

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 '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.

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.

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 '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.