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%I #30 Nov 14 2023 11:47:57
%S 1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,636,
%T 1272,25,50,1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,
%U 32768,636,1272,25,50,1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,636,1272,25,50
%N a(0) = 1; thereafter a(n) is obtained by doubling a(n-1) and repeatedly deleting any string of identical digits.
%C Periodic with period length 20.
%C Conjecture: If we start with any nonnegative number, and repeatedly double it and apply the "repeatedly delete any run of identical digits" operation described here, we eventually reach one of 0, 1, or 5.
%C In other words, the conjecture is that eventually we reach 0 or join the trajectory shown here or the trajectory shown in A323831.
%C The number of steps to reach 0, 1, or 5 is given in A323832.
%H Colin Barker, <a href="/A323830/b323830.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).
%F From _Colin Barker_, Feb 03 2019: (Start)
%F G.f.: (1 + 2*x)*(1 + 4*x^2 + 16*x^4 + 64*x^6 + 256*x^8 + 1024*x^10 + 4096*x^12 + 16384*x^14 + 636*x^16 + 25*x^18) / (1 - x^20).
%F a(n) = a(n-20) for n>19.
%F (End)
%F a(n+1) = A321801(2*a(n)). For general numbers, the "repeatedly delete any run of identical digits" operation corresponds to repeatedly applying A321801. - _Chai Wah Wu_, Feb 11 2019
%e After a(15) = 32768 we get 65536 which becomes 636 after deleting "55". Then doubling 636 we get 1272, then 2544 which becomes 25 after deleting "44", then 50, then 100 which becomes 1 after deleting "00", and now the sequence repeats.
%t dad[n_]:=FromDigits[FixedPoint[Flatten[Select[Split[#],Length[#]==1&]]&,IntegerDigits[2n]]];NestList[dad,1,100] (* _Paolo Xausa_, Nov 14 2023 *)
%o (PARI) Vec((1 + 2*x)*(1 + 4*x^2 + 16*x^4 + 64*x^6 + 256*x^8 + 1024*x^10 + 4096*x^12 + 16384*x^14 + 636*x^16 + 25*x^18) / (1 - x^20) + O(x^40)) \\ _Colin Barker_, Feb 03 2019
%Y Cf. A000079, A320487, A321801, A321802, A323831, A323832.
%Y See A035615 for a classic related base-2 sequence.
%K nonn,base,easy
%O 0,2
%A _N. J. A. Sloane_, Feb 03 2019, following a suggestion from _Yukun Yao_.
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