A288536 -id:A288536 - cp's OEIS frontend

A114611 Eventual period of the RATS sequence, where 0 indicates a divergent sequence.

0, 0, 8, 0, 0, 8, 0, 0, 2, 0, 0, 8, 0, 0, 8, 0, 0, 2, 0, 0, 8, 0, 0, 8, 0, 0, 2, 0, 18, 8, 0, 0, 8, 0, 0, 2, 0, 18, 8, 0, 0, 8, 0, 0, 2, 0, 18, 8, 18, 0, 8, 0, 0, 2, 0, 18, 8, 18, 0, 8, 0, 0, 2, 0, 18, 8, 18, 0, 2, 0, 0, 2, 0, 18, 8, 18, 0, 2, 0, 0, 2, 0, 18, 8, 18, 0, 2, 0, 0, 2, 0, 18, 8, 18, 0, 2

Offset: 1

Eric W. Weisstein, Dec 16 2005

a(A001651(n)) = 0; a(A114612(n)) = 2; a(A114613(n)) = 3; a(A114614(n)) = 8; a(A114615(n)) = 14; a(A114616(n)) = 18. - Reinhard Zumkeller, Mar 14 2012

Row 10 of A288537. - Andrey Zabolotskiy, Jun 14 2017

Eric Weisstein's World of Mathematics, RATS Sequence

Cf. A036839, A161593, A288536, A288537.

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.

Original entry on oeis.org

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

Offset: 2

Andrey Zabolotskiy, Jun 11 2017

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.

			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 ...

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, Conway’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.
Index entries for sequences related to RATS: Reverse, Add Then Sort

Cf. A004000, A036839, A114611 (row 10), A161593, A288535, A288536 (column 1).

A(2^t,1)=t.

A(3,3^A134067(p)-1)=p+3.

A288535 Array RATS(b,n) by upward antidiagonals: Reverse Add Then Sort digits of n>0 in base b>1.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 2, 1, 4, 3, 2, 4, 6, 8, 3, 2, 4, 6, 5, 4, 3, 2, 4, 1, 8, 10, 8, 7, 2, 4, 6, 8, 6, 15, 4, 3, 2, 4, 6, 8, 10, 12, 5, 14, 3, 2, 4, 6, 1, 10, 7, 18, 10, 4, 15, 2, 4, 6, 8, 10, 12, 14, 24, 15, 8, 3, 2, 4, 6, 8, 10, 12, 8, 21, 6, 5, 4, 15, 2, 4, 6

Offset: 2

Andrey Zabolotskiy, Jun 11 2017

			17 in base 3 is 122, 122+221=1120->112 which is 14 in decimal, thus RATS(3,17)=14.
The array begins:
1, 3, 3, 3,  3,  3,  7, ...
2, 4, 4, 8,  4,  8,  4, ...
2, 1, 6, 5, 10, 15,  5, ...
2, 4, 6, 8,  6, 12, 18, ...
2, 4, 1, 8, 10,  7, 14, ...

J. Thiel, On RATS sequences in general bases, Integers, 14 (2014), #A50.
Index entries for sequences related to RATS: Reverse, Add Then Sort

Cf. A004000, A036839 (row 10), A191780, A288536, A288537.

rats[n_, b_: 10] := FromDigits[Sort[IntegerDigits[n + FromDigits[Reverse[IntegerDigits[n, b]], b], b]], b];
Flatten[Table[rats[n, s + 2 - n], {s, 20}, {n, s}]]

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A114611 Eventual period of the RATS sequence, where 0 indicates a divergent sequence.

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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.

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A288535 Array RATS(b,n) by upward antidiagonals: Reverse Add Then Sort digits of n>0 in base b>1.

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