A036839 -id:A036839 - cp's OEIS frontend

A114614 Starting numbers for which the RATS sequence has eventual period 8.

3, 6, 12, 15, 21, 24, 30, 33, 39, 42, 48, 51, 57, 60, 66, 75, 84, 93, 102, 105, 111, 123, 129, 132, 138, 141, 147, 150, 159, 165, 168, 174, 177, 183, 186, 192, 195, 201, 204, 210, 219, 222, 228, 231, 237, 240, 246, 249, 258, 264, 267, 273, 276, 282, 285, 291

Offset: 1

Eric W. Weisstein, Dec 16 2005

A114611(a(n)) = 8. - Reinhard Zumkeller, Mar 14 2012

Eric Weisstein's World of Mathematics, RATS Sequence

Cf. A114611, A114612, A114613, A114614, A114615, A114616.

Cf. A066710, A036839.

A114615 Starting numbers for which the RATS sequence has eventual period 14.

Original entry on oeis.org

6999, 7089, 7179, 7269, 7359, 7449, 7539, 7629, 7719, 7809, 7998, 8088, 8178, 8268, 8358, 8448, 8538, 8628, 8718, 8808, 8997, 9087, 9177, 9267, 9357, 9447, 9537, 9627, 9699, 9717, 9789, 9807, 9879, 9969, 9996, 10128, 10167, 10185, 10191

Offset: 1

Eric W. Weisstein, Dec 16 2005

A114611(a(n)) = 14. - Reinhard Zumkeller, Mar 14 2012

Eric Weisstein's World of Mathematics, RATS Sequence

Cf. A114611, A114612, A114613, A114614, A114615, A114616.

Cf. A209879, A036839.

A114616 Starting numbers for which the RATS sequence has eventual period 18.

Original entry on oeis.org

29, 38, 47, 49, 56, 58, 65, 67, 74, 76, 83, 85, 92, 94, 110, 112, 118, 134, 137, 140, 142, 154, 155, 181, 187, 196, 209, 211, 217, 229, 233, 236, 239, 241, 253, 254, 259, 280, 286, 295, 299, 308, 310, 316, 319, 328, 329, 332, 335, 338, 340, 352, 353, 358

Offset: 1

Eric W. Weisstein, Dec 16 2005

A114611(a(n)) = 18. - Reinhard Zumkeller, Mar 14 2012

Eric Weisstein's World of Mathematics, RATS Sequence

Cf. A114611, A114612, A114613, A114614, A114615, A114616.

Cf. A209880, A036839.

A161593 Lengths of new periods in the RATS sequence (0 replacing infinity).

Original entry on oeis.org

0, 8, 2, 18, 2, 2, 2, 14, 2, 3, 2, 2, 2, 6

Offset: 1

J. H. Conway and Tanya Khovanova, Jun 14 2009

The values A114611(j) for those starting values j of the RATS mapping x->A036839(x) which end in cycles that cannot be reached starting from any smaller j.
Every integer > 1 appears in this sequence. - Andrey Zabolotskiy, Jun 11 2017
For other terms see Branicky link. - Michael S. Branicky, Dec 30 2022

			a(1)=A114611(0). a(2)=A114611(j=3)=8 with a cycle of length 8 shown in A066710.
A114611(j=6)=8 does not contribute because the cycle is the same as reached from j=3.
a(3)=A114611(9)=2 with a new cycle of length 2 shown in A066711.
A114611(j=12)=8 does not contribute because the cycle is the same as reached from j=3.
A114611(j=15)=8 does not contribute because 15->66->123 is the cycle as reached from j=3.
A114611(j=18)=2 does not contribute because the cycle is the same as reached from j=9.
A114611(j=21)=8 does not contribute because 21->33->66 reaches the same cycle as started from j=3.
a(4)=A114611(j=29)=18.

Michael S. Branicky, RATS Sequence Cycles.
Curtis Cooper, RATS.
Curtis Cooper and Robert E. Kennedy, Base 10 RATS Cycles and Arbitrarily Long Base 10 RATS Cycles, Applications of Fibonacci numbers, Vol. 8, Kluwer Acad. Publ., Dordrecht, 1999, pages 83-93.
Tanya Khovanova, Destinies of Numbers.

Cf. A004000, A036839, A066711, A066710, A288537.

Cf. A161590, A161592, A161596.

Comment and examples added by R. J. Mathar, Jul 07 2009

a(9)-a(14) from Michael S. Branicky, Dec 30 2022

A209878 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 20169.

Original entry on oeis.org

20169, 111267, 337788, 1122255, 4446666, 1111113, 2222244, 4446666, 1111113, 2222244, 4446666, 1111113, 2222244, 4446666, 1111113, 2222244, 4446666, 1111113, 2222244, 4446666, 1111113, 2222244, 4446666, 1111113, 2222244, 4446666, 1111113, 2222244, 4446666

Offset: 1

Reinhard Zumkeller, Mar 14 2012

A114613(1) = 20169 is the smallest starting number for a RATS trajectory leading to a cycle of length 3: A114611(20169) = 3;

a(n + 3) = a(n) for n > 4.

Eric Weisstein's World of Mathematics, RATS Sequence
Index entries for linear recurrences with constant coefficients, signature (0, 0, 1).

Cf. A004000, A066710, A066711, A209879, A209880.

a209878 n = a209878_list !! (n-1)
a209878_list = iterate a036839 20169

Join[{20169, 111267, 337788, 1122255},LinearRecurrence[{0, 0, 1},{4446666, 1111113, 2222244},25]] (* Ray Chandler, Aug 25 2015 *)

a(n + 1) = A036839(a(n)).

A209879 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 6999.

Original entry on oeis.org

6999, 15699, 11355, 66666, 123333, 445566, 111111, 222222, 444444, 888888, 1677777, 3455589, 11112333, 33444444, 77778888, 156666666, 123333378, 666669999, 1356666666, 123333789, 11111667, 22777788, 11115555, 66666666, 123333333, 445566666, 111122223

Offset: 1

Reinhard Zumkeller, Mar 14 2012

A114615(1) = 6999 is the smallest starting number for a RATS trajectory leading to a cycle of length 14: A114611(6999) = 14;

a(n + 14) = a(n) for n > 25.

Reinhard Zumkeller, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, RATS Sequence
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A004000, A066710, A209878, A066711, A209880.

a209879 n = a209879_list !! (n-1)
a209879_list = iterate a036839 6999

rats[n_]:=Module[{idnr=FromDigits[Reverse[IntegerDigits[n]]]}, FromDigits[ Sort[ IntegerDigits[idnr+n]]]]; NestList[rats,6999,30] (* Harvey P. Dale, May 29 2014 *)

a(n + 1) = A036839(a(n)).

A209880 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 29.

Original entry on oeis.org

29, 112, 233, 556, 1112, 2233, 5555, 1111, 2222, 4444, 8888, 16777, 34589, 112333, 444455, 889999, 1788899, 1177777, 4558889, 13444447, 77888888, 156667777, 233444489, 1112278888, 11999, 11119, 1223, 4444, 8888, 16777, 34589, 112333, 444455, 889999, 1788899

Offset: 1

Reinhard Zumkeller, Mar 14 2012

A114616(1) = 29 is the smallest starting number for a RATS trajectory leading to a cycle of length 18: A114611(29) = 18;

a(n + 18) = a(n) for n > 9.

Eric Weisstein's World of Mathematics, RATS Sequence
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A004000, A066710, A209878, A066711, A209879.

a209880 n = a209880_list !! (n-1)
a209880_list = iterate a036839 29

NestList[FromDigits[Sort[IntegerDigits[#+IntegerReverse[#]]]]&,29,40] (* or *) PadRight[{29,112,233,556,1112,2233,5555,1111,2222},50,{4558889,13444447,77888888,156667777,233444489,1112278888,11999,11119,1223,4444,8888,16777,34589,112333,444455,889999,1788899,1177777}] (* Harvey P. Dale, Sep 17 2018 *)

a(n + 1) = A036839(a(n)).

A161596 Numbers in cycles of RATS sequences.

Original entry on oeis.org

78, 111, 117, 156, 222, 288, 444, 888, 1223, 1677, 3489, 4444, 8888, 11119, 11127, 11667, 11999, 12333, 16777, 23388, 27888, 34589, 44556, 111177, 112333, 228888, 444455, 889999, 1111113, 1177777, 1788899, 2222244, 4446666, 4558889, 11144445, 13444447, 55556688

Offset: 1

J. H. Conway and Tanya Khovanova, Jun 14 2009

The set of all numbers in any cycle of RATS sequences, sorted into natural order.
This implies that for any value a(j) in this sequence, A036839(a(j)) is again member of the sequence.
See Branicky link for larger terms. - Michael S. Branicky, Dec 30 2022

			The numbers 111, 222, 444, 888, 1677, 3489, 12333 and 44556 are in the sequence because they are in the cycle shown in A066710. The numbers 117 and 288 are in the cycle demonstrated in A066711.
The numbers 4444, 8888, 16777, 34589, 112333, 444455, ..., 1112278888, 11999, 1119, 1223 are in the cycle started at A161590(4). The numbers 11127 and 23388 are in the cycle started at A161590(7).

Michael S. Branicky, RATS Sequence Cycles.

Cf. A004000, A036839, A066711, A066710, A161590, A161592.

Descriptive comment and examples added by R. J. Mathar, Jul 08 2009

a(20) and beyond from Michael S. Branicky, Dec 30 2022

A161592 Except for the first term the number in the sequence is the smallest number in a new cycle of a RATS sequence with a new destiny. The first term is the best analog of this for the "infinite cycle".

Original entry on oeis.org

12334444, 111, 117, 1223, 78, 111177, 11127, 11144445, 11667, 1111113

Offset: 1

J. H. Conway & Tanya Khovanova, Jun 14 2009

"Destiny" means the smallest element of the cycle that the trajectory ends up in.

All seeds except those generating the cycles listed here produce an open non-cyclic family (thus without lowest element) but with a regular structure like 12334444, 55667777, 123334444, 556667777, 1233334444, 5566667777,..., and with an arbitrary start-up like 1, 2, 4, 8, 16, 77, 145, 668, 1345, 6677, 13444, 55778, 133345, 666677, 1333444, 5567777, 12333445, 66666677, 133333444, 556667777, 1233334444, 5566667777, 12333334444, 55666667777, 123333334444, ... Notice that here we fall into the regular regime starting with 1233334444 (four threes). The sequence gives 12-(two threes)-4444 as a representative with index 1. - Wouter Meeussen, Jul 26 2009

Tanya Khovanova, Destinies of Numbers - _Tanya Khovanova_, Aug 10 2009

Cf. A004000, A036839, A066710, A066711, A161590.

Cf. A161593, A161596. - Tanya Khovanova, Aug 10 2009

11667, 1111113 from Wouter Meeussen, Jul 04 2009

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.

cp's OEIS Frontend

A114614 Starting numbers for which the RATS sequence has eventual period 8.

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A114615 Starting numbers for which the RATS sequence has eventual period 14.

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A114616 Starting numbers for which the RATS sequence has eventual period 18.

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A161593 Lengths of new periods in the RATS sequence (0 replacing infinity).

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A209878 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 20169.

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A209879 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 6999.

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A209880 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 29.

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A161596 Numbers in cycles of RATS sequences.

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A161592 Except for the first term the number in the sequence is the smallest number in a new cycle of a RATS sequence with a new destiny. The first term is the best analog of this for the "infinite cycle".

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