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A077076 Trajectory of 537 under the Reverse and Add! operation carried out in base 2, written in base 10.

%I A077076 #14 Sep 08 2022 08:45:07
%S A077076 537,1146,1899,3618,4713,9522,14427,28386,37533,84966,138123,353004,
%T A077076 466209,738024,833301,1525224,1718853,3048912,3239469,6196176,6583437,
%U A077076 12389280,12770397,24975264,25749789,49944384,50706621,100282176
%N A077076 Trajectory of 537 under the Reverse and Add! operation carried out in base 2, written in base 10.
%C A077076 The base 2 trajectory of 537 = A075252(4) provably does not contain a palindrome. A proof can be based on the formula given below.
%C A077076 lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 2 = 1.
%C A077076 lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 2 = 0.
%C A077076 Interleaving of 3*A177682, 6*A177683, 3*A177684, 6*A177685.
%H A077076 Reinhard Zumkeller, <a href="/A077076/b077076.txt">Table of n, a(n) for n = 0..1000</a>
%H A077076 Klaus Brockhaus, <a href="/A058042/a058042.txt">On the 'Reverse and Add!' algorithm in base 2</a>
%H A077076 <a href="/index/Res#RAA">Index entries for sequences related to Reverse and Add!</a>
%F A077076 a(0), ..., a(11) as above; for n > 11 and
%F A077076 n = 0 (mod 4): a(n) = 3*2^(2*k+13)+18249*2^k-3 where k = (n-4)/4;
%F A077076 n = 1 (mod 4): a(n) = 6*2^(2*k+13)-12102*2^k where k = (n-5)/4;
%F A077076 n = 2 (mod 4): a(n) = 6*2^(2*k+13)+11718*2^k-3 where k = (n-6)/4;
%F A077076 n = 3 (mod 4): a(n) = 12*2^(2*k+13)-11910*2^k where k = (n-7)/4.
%F A077076 G.f.: 3*(179+382*x+96*x^2+60*x^3-328*x^4-444*x^5+1170*x^6+2232*x^7 +1166*x^8+5644*x^9+15402*x^10+46922*x^11+39850*x^12-62920*x^13-132612*x^14 -97532*x^15-34148*x^16+83800*x^17+109224*x^18+21856*x^19) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)).
%F A077076 G.f. for the sequence starting at a(12): 3*x^12*(155403+246008*x-188442*x^2-229616*x^3-260350*x^4-508920*x^5+293388*x^6+492528*x^7) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4))
%F A077076 a(n+1) = A055944(a(n)). - _Reinhard Zumkeller_, Apr 21 2013
%e A077076 537 (decimal) = 1000011001 -> 1000011001 + 1001100001 = 10001111010= 1146 (decimal).
%t A077076 NestWhileList[# + IntegerReverse[#, 2] &, 537,  # !=
%t A077076 IntegerReverse[#, 2] &, 1, 27] (* _Robert Price_, Oct 18 2019 *)
%o A077076 (PARI) trajectory(n,steps) = {local(v, k=n); for(j=0, steps, print1(k, ", "); v=binary(k); k+=sum(j=1, #v, 2^(j-1)*v[j]))};
%o A077076 trajectory(537,27);
%o A077076 (Magma) trajectory:=function(init, steps, base) a:=init; S:=[a]; for n in [1..steps] do a+:=Seqint(Reverse(Intseq(a,base)),base); Append(~S, a); end for; return S; end function; trajectory(537, 27, 2);
%o A077076 (Haskell)
%o A077076 a077076 n = a077076_list !! n
%o A077076 a077076_list = iterate a055944 537  -- _Reinhard Zumkeller_, Apr 21 2013
%Y A077076 Cf. A058042 (trajectory of 22 in base 2, written in base 2), A061561 (trajectory of 22 in base 2), A075253 (trajectory of 77 in base 2), A075268 (trajectory of 442 in base 2), A077077 (trajectory of 775 in base 2), A075252 (trajectory of n in base 2 does not reach a palindrome and (presumably) does not join the trajectory of any term m < n).
%Y A077076 Cf. A177682 (a(4*n)/3), A177683 (a(4*n+1)/6), A177684 (a(4*n+2)/3), A177685 (a(4*n+3)/6).
%K A077076 base,nonn
%O A077076 0,1
%A A077076 _Klaus Brockhaus_, Oct 25 2002
%E A077076 Comment edited and three comments added, g.f. edited, PARI program revised, MAGMA program and crossrefs added by _Klaus Brockhaus_, May 12 2010