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A164338 Conway's creeper sequence.

%I A164338 #17 Jul 22 2025 07:15:57
%S A164338 12334444,55667777,123334444,556667777,1233334444,5566667777,
%T A164338 12333334444,55666667777,123333334444,556666667777,1233333334444,
%U A164338 5566666667777,12333333334444,55666666667777,123333333334444
%N A164338 Conway's creeper sequence.
%C A164338 Trajectory of 12334444 under the RATS function A036839.
%C A164338 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.
%C A164338 Continues with the obvious digital pattern.
%C A164338 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.
%H A164338 Reinhard Zumkeller, <a href="/A164338/b164338.txt">Table of n, a(n) for n = 1..100</a>
%H A164338 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RATSSequence.html">RATS Sequence</a>
%H A164338 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,11,0,-10).
%F A164338 a(n+2) = 10 a(n) - 9996 (n odd)
%F A164338 a(n+2) = 10 a(n) - 9993 (n even)
%F A164338 a(n+4) = 11 a(n+2) - 10 a(n)
%F A164338 a(n + 1) = A036839(a(n)). [_Reinhard Zumkeller_, Mar 14 2012]
%F A164338 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
%o A164338 (Haskell)
%o A164338 a164338 n = a164338_list !! (n-1)
%o A164338 a164338_list = iterate a036839 12334444
%o A164338 -- _Reinhard Zumkeller_, Mar 14 2012
%Y A164338 Cf. A036839 (RATS function), A161590, A161592, A161593.
%Y A164338 Cf. A114611, A114612.
%K A164338 base,easy,nonn
%O A164338 1,1
%A A164338 _David W. Wilson_, Aug 13 2009