A114611 -id:A114611 - cp's OEIS frontend

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

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)).

A161590 Initial value x of a RATS trajectory x->A036839(x) ending in a cycle unreachable by any smaller initial value.

Original entry on oeis.org

1, 3, 9, 29, 69, 2079, 3999, 6999, 10677, 20169, 10049598, 20008989, 100014888, 100074268

Offset: 1

J. H. Conway and Tanya Khovanova, Jun 14 2009, Jul 04 2009

This is one way of book-keeping of new "destinies" (the smallest element of the cycle that the trajectory ends up in).
The value 1 is a placeholder for all non-cyclic trajectories.
Next terms are respectively <= 10000122228, 20000666679, 2000001113379, 2000001113559, 9999999999999, 100000044444447.  See Branicky link for further upper bounds. - Michael S. Branicky, Dec 30 2022

			The RATS (Reverse Add Then Sort) algorithm applied to 69 produces a sequence 69, 156, 78, 156, 78, ...
Its cycle {156, 78} appears not if the algorithm is started with any number in the range 0 to 68, so 69 is added to the sequence.

Michael S. Branicky, RATS Sequence Cycles.
Tanya Khovanova, Destinies of Numbers. [From _Tanya Khovanova_, Aug 10 2009]

Cf. A036839, A066710, A066711, A004000, A114611, A161592, A161593, A161596.

10677, 20169 from Wouter Meeussen, Jul 04 2009
Definition rephrased by R. J. Mathar, Jul 08 2009
a(11)-a(14) from Michael S. Branicky, Dec 30 2022

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.

A164338 Conway's creeper sequence.

Original entry on oeis.org

12334444, 55667777, 123334444, 556667777, 1233334444, 5566667777, 12333334444, 55666667777, 123333334444, 556666667777, 1233333334444, 5566666667777, 12333333334444, 55666666667777, 123333333334444

Offset: 1

David W. Wilson, Aug 13 2009

Trajectory of 12334444 under the RATS function A036839.
John Conway calls this sequence "the creeper" and conjectures that the RATS trajectory of every n >= 1 eventually enters a cycle or the creeper. David Wilson confirms this conjecture for n <= 10^10.
Continues with the obvious digital pattern.
Since a(n+2) = a(n) except for an added digit, this sequence can be described as a quasi-cycle of period 2 with smallest element 12334444. This is how it is treated in related sequences such as A161590, A161592 and A161593.

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 (0,11,0,-10).

Cf. A036839 (RATS function), A161590, A161592, A161593.

Cf. A114611, A114612.

a164338 n = a164338_list !! (n-1)
a164338_list = iterate a036839 12334444
-- Reinhard Zumkeller, Mar 14 2012

a(n+2) = 10 a(n) - 9996 (n odd)
a(n+2) = 10 a(n) - 9993 (n even)
a(n+4) = 11 a(n+2) - 10 a(n)
a(n + 1) = A036839(a(n)). [Reinhard Zumkeller, Mar 14 2012]
G.f.: x*(-55677770*x^3 - 12344440*x^2 + 55667777*x + 12334444)/(10*x^4 - 11*x^2 + 1). - Chai Wah Wu, Feb 08 2020

A288536 The eventual period of the RATS sequence in base n starting from 1; 0 is for infinity.

Original entry on oeis.org

1, 3, 2, 2, 8, 4, 3, 2, 0, 28, 90, 8, 72, 3, 4, 2, 64, 0, 18, 4, 18, 20, 396, 8, 160, 120, 18, 6, 28, 4, 5, 2, 210, 384, 240, 0, 648, 1242, 240, 4, 660, 18, 798, 380, 852, 1298, 1771, 8, 0, 160, 16, 372, 520, 1404, 1740, 6, 36, 2072, 1856, 380, 300, 215, 6, 2, 3384, 50, 2310, 3784, 2904

Offset: 2

Andrey Zabolotskiy, Jun 11 2017

Eventual period of 1 under the mapping x->A288535(n,x), or 0 if there is a divergence and thus no eventual period.
Column 1 of A288537.
In Thiel's terms, the zeroes a(10), a(19), and a(37) correspond to quasiperiodic divergent RATS sequences with quasiperiod 2, while a(50)=0 corresponds to a sequence with quasiperiod 3.

			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)=3.

S. Shattuck and C. Cooper, Divergent RATS sequences, Fibonacci Quart., 39 (2001), 101-106.
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, A114611, A288535, A288537, A237671, A072137.

A066713 RATS(2^n): Reverse Add the digits of 2^n, Then Sort: a(n) = A036839(2^n).

Original entry on oeis.org

2, 4, 8, 16, 77, 55, 11, 499, 89, 277, 2255, 145, 11, 1111, 44567, 111499, 12299, 1234, 3467, 113467, 677789, 144556, 1222889, 14445667, 4577789, 55669999, 1134899, 11356999, 12237899, 445557799, 1223555555, 11113366, 1122222266

Offset: 0

N. J. A. Sloane, Jan 19 2002

A114611(a(n)) = 0, as A114611(A000079(n)) = 0. - Reinhard Zumkeller, Mar 14 2012

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, RATS Sequence

See A004000, A036839 for more information.

a066713 = a036839 . (2 ^)  -- Reinhard Zumkeller, Mar 14 2012

def A066713(n):
    m = 2**n
    return int(''.join(sorted(str(m+int(str(m)[::-1]))))) # Chai Wah Wu, Feb 07 2020

cp's OEIS Frontend

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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A161590 Initial value x of a RATS trajectory x->A036839(x) ending in a cycle unreachable by any smaller initial value.

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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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A164338 Conway's creeper sequence.

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A288536 The eventual period of the RATS sequence in base n starting from 1; 0 is for infinity.

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A066713 RATS(2^n): Reverse Add the digits of 2^n, Then Sort: a(n) = A036839(2^n).

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