This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A288537 #10 Feb 16 2025 08:33:47 %S A288537 1,3,1,2,3,1,2,2,3,1,8,2,2,3,1,4,8,2,2,3,1,3,4,8,2,2,3,1,2,3,2,8,2,2, %T A288537 3,1,0,2,3,4,2,2,2,3,1,28,0,2,3,4,8,2,2,3,1,90,28,8,2,6,2,8,2,2,3,1,8, %U A288537 90,28,0,2,3,4,8,2,2,3,1,72,8,90,28,0,2 %N A288537 Array A(b,n) by upward antidiagonals (b>1, n>0): the eventual period of the RATS sequence in base b starting from n; 0 is for infinity. %C A288537 Eventual period of n under the mapping x->A288535(b,x), or 0 if there is a divergence and thus no eventual period. %C A288537 For b = 3*2^m - 2 with m>1, row b contains all sufficiently large even integers if m is odd, or just all sufficiently large integers if m is even. %C A288537 For b = 1 or 10 (mod 18) or b = 1 (mod (2^q-1)^2) with q>2, there are 0's in row b. %C A288537 Conway conjectured that in row (base) 10, all 0's correspond to the same divergent RATS sequence called the Creeper (A164338). In Thiel's terms, it is quasiperiodic with quasiperiod 2, i.e., after every 2 steps the number of one of the digits (in this case, 3 or 6) increases by 1 while other digits stay unchanged. In other bases, 0's may correspond to different divergent RATS sequences. Thiel conjectured that the divergent RATS sequences are always quasiperiodic. %H A288537 Curtis Cooper, <a href="http://cs.ucmo.edu/~cnc8851/rats.html">RATS</a>. %H A288537 R. K. Guy, <a href="https://doi.org/10.2307/2325149">Conway's RATS and other reversals</a>, Amer. Math. Monthly, 96 (1989), 425-428. %H A288537 S. Shattuck and C. Cooper, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/39-2/shattuck.pdf">Divergent RATS sequences</a>, Fibonacci Quart., 39 (2001), 101-106. %H A288537 J. Thiel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Thiel/thiel2.pdf">Conway’s RATS Sequences in Base 3</a>, Journal of Integer Sequences, 15 (2012), Article 12.9.2. %H A288537 J. Thiel, <a href="https://www.emis.de/journals/INTEGERS/papers/o50/o50.pdf">On RATS sequences in general bases</a>, Integers, 14 (2014), #A50. %H A288537 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RATSSequence.html">RATS Sequence</a>. %H A288537 <a href="/index/Ra#RATS">Index entries for sequences related to RATS: Reverse, Add Then Sort</a> %F A288537 A(2^t,1)=t. %F A288537 A(3,3^A134067(p)-1)=p+3. %e A288537 In base 3, the RATS mapping acts as 1 -> 2 -> 4 (11 in base 3) -> 8 (22 in base 3) -> 13 (112 in base 3) -> 4, which has already been seen 3 steps ago, so A(3,1)=3. %e A288537 The array begins: %e A288537 1, 1, 1, 1, 1, 1, ... %e A288537 3, 3, 3, 3, 3, 3, ... %e A288537 2, 2, 2, 2, 2, 2, ... %e A288537 2, 2, 2, 2, 2, 2, ... %e A288537 8, 8, 8, 8, 2, 8, ... %e A288537 4, 4, 2, 4, 4, 2, ... %e A288537 3, 3, 3, 3, 6, 3, ... %e A288537 2, 2, 2, 2, 2, 2, ... %e A288537 0, 0, 8, 0, 0, 8, ... %e A288537 28, 28, 28, 28, 2, 28, ... %e A288537 90, 90, 90, 90, 90, 90 ... %Y A288537 Cf. A004000, A036839, A114611 (row 10), A161593, A288535, A288536 (column 1). %K A288537 nonn,tabl,base %O A288537 2,2 %A A288537 _Andrey Zabolotskiy_, Jun 11 2017