A114612 -id:A114612 - cp's OEIS frontend

A004000 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 1.

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, 556666667777, 1233333334444, 5566666667777, 12333333334444

Offset: 1

N. J. A. Sloane

It is conjectured that no matter what the starting term is, repeatedly applying RATS leads either to this sequence or into a cycle of finite length, such as those in A066710 and A066711.

			668 -> 668 + 866 = 1534 -> 1345.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..2002 (first 200 terms from T. D. Noe)
R. K. Guy, Conway's RATS and other reversals, Amer. Math. Monthly, 96 (1989), 425-428.
Eric Weisstein's World of Mathematics, RATS Sequence.

Cf. A036839, A066710, A066711, A066713, A164338, A161593, A114611, A114612, A209878, A209879, A209880.

a004000_list = iterate a036839 1  -- Reinhard Zumkeller, Mar 14 2012

[ n eq 1 select 1 else Seqint(Reverse(Sort(Intseq(p + Seqint(Reverse(Intseq(p))) where p is Self(n-1))))) : n in [1..10]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 20061

read transforms; RATS := n -> digsort(n + digrev(n)); b := [1]; t := [1]; for n from 1 to 50 do t := RATS(t); b := [op(b),t]; od: b;

NestList[FromDigits[Sort[IntegerDigits[#+FromDigits[Reverse[ IntegerDigits[#]]]]]]&,1,30] (* Harvey P. Dale, Nov 29 2011 *)

step(n)=fromdigits(vecsort(digits(n+fromdigits(Vecrev(digits(n)))))) \\ Charles R Greathouse IV, Jun 23 2017

l = [0, 1]
for n in range(2, 51):
    x = str(l[n - 1])
    l.append(int(''.join(sorted(str(int(x) + int(x[::-1]))))))
print(l[1:]) # Indranil Ghosh, Jul 05 2017

Let a(n) = k, form m by Reversing the digits of k, Add m to k Then Sort the digits of the sum into increasing order to get a(n+1).
a(n+1) = A036839(a(n)). - Reinhard Zumkeller, Mar 14 2012
A010888(a(n)) = A153130(n-1). - Ivan N. Ianakiev, Nov 27 2014
a(2n-1) = (37 * 10^(n-3) + 3332)/3, n >= 11; a(2n) = (167 * 10^(n-3) + 3331)/3, n >= 10. - Jianing Song, May 06 2021

Entry revised by N. J. A. Sloane, Jan 19 2002

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

Original entry on oeis.org

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.

A066711 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 9.

Original entry on oeis.org

9, 18, 99, 189, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117, 288, 117

Offset: 1

N. J. A. Sloane, Jan 19 2002

a(1) = A114612(1) = 9; A114611(3) = 2. - Reinhard Zumkeller, Mar 14 2012

			668 -> 668 + 866 = 1534 -> 1345.

R. K. Guy, Conway's RATS and other reversals, Amer. Math. Monthly, 96 (1989), 425-428.
J. Thiel, Conway’s RATS Sequences in Base 3, Journal of Integer Sequences, 15 (2012), #12.9.2. - _N. J. A. Sloane_, Jan 02 2013
Eric Weisstein's World of Mathematics, RATS Sequence
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A004000, A036839, A066710, A033896, A114611, A114612.

Cf. A004000, A066710, A209878, A209879, A209880.

a066711_list = iterate a036839 9  -- Reinhard Zumkeller, Mar 14 2012

NestList[ FromDigits[ Sort[ IntegerDigits[# + FromDigits[ Reverse[ IntegerDigits[#]]]]]] &, 9, 48] (* Jayanta Basu, Aug 13 2013 *)
Join[{9, 18, 99, 189},LinearRecurrence[{0, 1},{117, 288},45]] (* Ray Chandler, Aug 25 2015 *)

from itertools import accumulate
def rats(anm1, _):
    return int("".join(sorted(str(anm1 + int(str(anm1)[::-1])))))
print(list(accumulate([9]*49, rats))) # Michael S. Branicky, Sep 18 2021

Let a(n) = k, form m by Reversing the digits of k, Add m to k Then Sort the digits of the sum into increasing order to get a(n+1).
Periodic with period 2.
a(n+1) = A036839(a(n)). - Reinhard Zumkeller, Mar 14 2012
G.f.: x*(-99*x^5 - 18*x^4 - 171*x^3 - 90*x^2 - 18*x - 9)/(x^2 - 1). - Chai Wah Wu, Feb 07 2020

A114613 Starting numbers for which the RATS sequence has eventual period 3.

Original entry on oeis.org

20169, 20709, 21159, 22149, 23139, 24129, 25119, 26109, 27099, 28089, 29079, 30159, 30168, 30708, 30789, 31149, 31158, 31779, 32139, 32148, 32769, 33129, 33138, 33759, 34119, 34128, 34749, 35109, 35118, 35739, 36108, 36729, 37098

Offset: 1

Eric W. Weisstein, Dec 16 2005

A114611(a(n)) = 3. - Reinhard Zumkeller, Mar 14 2012

Eric Weisstein's World of Mathematics, RATS Sequence

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

Cf. A209878, A036839.

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

Original entry on oeis.org

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.

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

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A004000 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 1.

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

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A066711 RATS: Reverse Add Then Sort the digits applied to previous term, starting with 9.

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A114613 Starting numbers for which the RATS sequence has eventual period 3.

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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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A164338 Conway's creeper sequence.

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