So far I see all pairs (a,a) occurring [except (1,1)] reasonably early (I've checked through a=1023). The longest one of those took to appear was 992, which occurred at 662,150,300. The pairs that seem to be taking the longest to appear are (0,a); after this, they are values (a,b) where either a or b is in the range 4..10, or where b-a is 5 or 6. The main thing I'm seeing is that the sequence is a series of ramps; a ramp starts with 0 and ends with a previously unseen value. Between these two values is a series of usually increasing numbers that appear to increase quickly. The length of the ramps varies but the average ramp length increases slowly. Most of the small integers that recur are the ramp length. If you look at the frequency of occurrence of values, you see a graph with some interesting and significant undulations. As the sequence continues those undulations move to the right, but the overall shape of the undulations appears pretty self-similar. I'll try to get some graphs up soon. Some late-appearing pairs where (a,b) are both less than 1024 are: See 0 34 at 250327022558 See 943 4 at 250470176885 See 9 937 at 250505969939 See 9 949 at 251219693325 See 924 4 at 251411817200 See 9 962 at 252043560162 See 10 708 at 252079357435 See 9 950 at 252247778126 See 577 583 at 252730372074 See 1 875 at 253071570424 See 481 5 at 254277582179 See 751 756 at 254389321902 See 841 4 at 254643579836 Frequences of occurrences; see the undulations? 0 23715597276 1 3858369548 2 1292639191 3 89031007 4 8625117 5 2302753 6 4589090 7 155877539 8 1847170967 9 6193152432 10 7958563024 11 6225475874 12 4061764482 13 2196264487 14 981661792 15 420357422 16 176297472 17 196811670 18 185357188 19 753266367 20 767663426 21 1032462039 22 870293065 23 485252403 24 382523736 25 120254552 26 109793239 27 145701135 28 167362679 29 317189883 30 619791368 31 648210091 32 672215340 33 577231661 34 462054056 35 283492644 36 173878733 37 146474139 38 165726872 39 234593490 40 318784422 41 408898611 42 448076370 On Sun, Jun 16, 2019 at 12:31 PM Tom Karzes <karzes@sonic.net> wrote:
Is anything known about the occurrences of consecutive pairs of the same number in the series? The pair 0,0 obviously occurs once, at the very beginning. And the pair 1,1 cannot occur, since it would produce an infinite repeating pattern. Other pairs do occur though (I believe 2, 3, 8, etc.)
Tom
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