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Occurences of a certain factor in the Kolakoski word

16 février 2011 | Catégories: Kolakoski, sage | View Comments

The factor 212212112112212112122112112212212112112212112122112112 occurs in the Kolakoski word at the positions 3190, 6366, 7614, 12950, 13765, 14277, 20385, 21344, 39674 and so on. The successive gaps of these first eight occurences are 3176, 1248, 5336, 815, 512, 6108, 959, 18330. Are the gap between all of these occurences bounded or not? The following table lists only the gap that are going increasingly for the Kolakoski word up to 100 billion (=10^9).

i ith occurence (i+1)th occurence gap
0 3190 6366 3176
2 7614 12950 5336
5 14277 20385 6108
7 21344 39674 18330
8 39674 58946 19272
17 85963 107018 21055
225 945732 966924 21192
685 2832810 2858670 25860
822 3549762 3583161 33399
965 4188934 4223349 34415
1526 6773992 6809208 35216
7117 31279355 31320896 41541
18054 79167173 79213333 46160
20838 90761663 90809563 47900
176198 768146935 768195609 48674
292643 1278852293 1278906094 53801
554207 2424441033 2424495471 54438
590128 2581311276 2581367730 56454
948506 4150204101 4150264753 60652
1156072 5055131250 5055199553 68303
1514374 6620302433 6620372047 69614
11154155 48788048239 48788121823 73584

The following image draws all the gaps in function of i for the first 10 million digit.

/Files/2011/kolakoski_from0_to10000000.png
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