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 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. 4
 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, 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, 90, 28, 0, 2, 3, 4, 8, 2, 2, 3, 1, 72, 8, 90, 28, 0, 2 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 COMMENTS Eventual period of n under the mapping x->A288535(b,x), or 0 if there is a divergence and thus no eventual period. 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. 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. 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. LINKS Curtis Cooper, RATS. R. K. Guy, Conway's RATS and other reversals, Amer. Math. Monthly, 96 (1989), 425-428. S. Shattuck and C. Cooper, Divergent RATS sequences, Fibonacci Quart., 39 (2001), 101-106. J. Thiel, Conwayâ€™s RATS Sequences in Base 3, Journal of Integer Sequences, 15 (2012), Article 12.9.2. J. Thiel, On RATS sequences in general bases, Integers, 14 (2014), #A50. Eric Weisstein's World of Mathematics, RATS Sequence. FORMULA A(2^t,1)=t. A(3,3^A134067(p)-1)=p+3. EXAMPLE 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. The array begins: 1, 1, 1, 1, 1, 1, ... 3, 3, 3, 3, 3, 3, ... 2, 2, 2, 2, 2, 2, ... 2, 2, 2, 2, 2, 2, ... 8, 8, 8, 8, 2, 8, ... 4, 4, 2, 4, 4, 2, ... 3, 3, 3, 3, 6, 3, ... 2, 2, 2, 2, 2, 2, ... 0, 0, 8, 0, 0, 8, ... 28, 28, 28, 28, 2, 28, ... 90, 90, 90, 90, 90, 90 ... CROSSREFS Cf. A004000, A036839, A114611 (row 10), A161593, A288535, A288536 (column 1). Sequence in context: A004591 A195588 A153510 * A167373 A079722 A079723 Adjacent sequences:  A288534 A288535 A288536 * A288538 A288539 A288540 KEYWORD nonn,tabl,base AUTHOR Andrey Zabolotskiy, Jun 11 2017 STATUS approved

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Last modified August 26 01:47 EDT 2019. Contains 326324 sequences. (Running on oeis4.)