A036839 -id:A036839 - cp's OEIS frontend

A164338 Conway's creeper sequence.

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

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}]]

A275218 Numbers in 2-cycles of RATS sequences.

Original entry on oeis.org

78, 117, 156, 288, 11127, 11667, 23388, 27888, 111177, 228888, 111111777, 222888888, 1111122267, 3333337788, 111111117777, 222288888888, 111111111177777, 222228888888888, 111111111111777777, 222222888888888888

Offset: 1

Robert Israel, Jul 20 2016

Numbers n such that A036839(A036839(n)) = n.
Subset of A161596.
Contains A002275(3*k) + 6*A002275(k) and 2*A002275(3*k)+6*A002275(2*k) for all k>0.
In particular, this sequence and A161596 are infinite.
Do all sufficiently large members of the sequence have the form A002275(3*k) + 6*A002275(k) or 2*A002275(3*k)+6*A002275(2*k)?

			78 is in the sequence because A036839(78) = 156 and A036839(156) = 78.

Cf. A002275, A036839, A161596.

rev:= proc(n) local t,L;
   L:= convert(n,base,10);
   add(10^j*L[-1-j],j=0..nops(L)-1)
end proc:
sord:= proc(n) local L,t;
  L:= sort(convert(n,base,10),`>`);
  add(10^j*L[1+j],j=0..nops(L)-1)
end proc:
rats:= proc(n) option remember;  sord(n + rev(n)) end proc:
Res:= NULL:
for d from 1 to 15 do
  for x1 from 0 to d do
    for x2 from 0 to d-x1 do
      for x3 from 0 to d-x1-x2 do
         for x4 from 0 to d-x1-x2-x3 do
           for x5 from 0 to d-x1-x2-x3-x4 do
             for x6 from 0 to d-x1-x2-x3-x4-x5 do
               for x7 from 0 to d-x1-x2-x3-x4-x5-x6 do
                 for x8 from 0 to d-x1-x2-x3-x4-x5-x6-x7 do
                   x9:= d-x1-x2-x3-x4-x5-x6-x7-x8;
                   L:= [1$x1,2$x2,3$x3,4$x4,5$x5,6$x6,7$x7,8$x8,9$x9];
                   x:= add(L[-i]*10^(i-1),i=1..d);
                   if rats(rats(x)) = x then Res:= Res,x fi
od od od od od od od od od:
sort([Res]);

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A164338 Conway's creeper sequence.

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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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A275218 Numbers in 2-cycles of RATS sequences.

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