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A061561 Trajectory of 22 under the Reverse and Add! operation carried out in base 2.

%I A061561 #22 Sep 08 2022 08:45:03
%S A061561 22,35,84,105,180,225,360,405,744,837,1488,1581,3024,3213,6048,6237,
%T A061561 12192,12573,24384,24765,48960,49725,97920,98685,196224,197757,392448,
%U A061561 393981,785664,788733,1571328,1574397,3144192,3150333,6288384,6294525
%N A061561 Trajectory of 22 under the Reverse and Add! operation carried out in base 2.
%C A061561 Sequence A058042 written in base 10. 22 is the smallest number whose base 2 trajectory does not contain a palindrome.
%C A061561 lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 2 = 0.
%C A061561 lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 2 = 1. - _Klaus Brockhaus_, Dec 09 2009
%H A061561 T. D. Noe, <a href="/A061561/b061561.txt">Table of n, a(n) for n=0..500</a>
%H A061561 Klaus Brockhaus, <a href="/A058042/a058042.txt">On the 'Reverse and Add!' algorithm in base 2</a>
%H A061561 <a href="/index/Res#RAA">Index entries for sequences related to Reverse and Add!</a>
%F A061561 a(0) = 22; a(1) = 35; for n > 1 and n = 2 (mod 4): a(n) = 6*2^(2*k)-3*2^k where k = (n+6)/4; n = 3 (mod 4): a(n) = 6*2^(2*k)+3*2^k-3 where k = (n+5)/4; n = 0 (mod 4): a(n) = 12*2^(2*k)-3*2^k where k = (n+4)/4; n = 1 (mod 4): a(n) = 12*2^(2*k)+9*2^k-3 where k = (n+3)/4. [_Klaus Brockhaus_, Sep 05 2002]
%F A061561 G.f.: (22+35*x+18*x^2-72*x^4-90*x^5-48*x^6-60*x^7+80*x^8+112*x^9) / ((1-x)*(1+x)*(1-2*x^2)*(1-2*x^4)). [_Klaus Brockhaus_, Sep 05 2002, edited Dec 09 2009]
%F A061561 a(n+1) = A055944(a(n)). - _Reinhard Zumkeller_, Apr 21 2013
%t A061561 binRA[n_] := If[Reverse[IntegerDigits[n, 2]] == IntegerDigits[n, 2], n, FromDigits[Reverse[IntegerDigits[n, 2]], 2] + n]; NestList[binRA, 22, 100] (* Adapted from _Ben Branman_'s code for A213012, _Alonso del Arte_, Jun 02 2012 *)
%o A061561 (ARIBAS) m := 22; stop := 36; c := 0; while c < stop do write(m,","); k := bit_length(m); rev := 0; for i := 0 to k-1 do if bit_test(m,i) then rev := bit_set(rev,k-1-i); end; end; inc(c); m := m+rev; end;.
%o A061561 (PARI) {m=22; stop=36; c=0; while(c<stop,print1(k=m,","); rev=0; while(k>0,d=divrem(k,2); k=d[1]; rev=2*rev+d[2]); c++; m=m+rev)}
%o A061561 (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(22, 35, 2); // _Klaus Brockhaus_, Dec 09 2009
%o A061561 (Haskell)
%o A061561 a061561 n = a061561_list !! n
%o A061561 a061561_list = iterate a055944 22  -- _Reinhard Zumkeller_, Apr 21 2013
%Y A061561 Cf. A035522 (trajectory of 1 in base 2), A058042 (trajectory of 22 in base 2, written in base 2), A075253 (trajectory of 77 in base 2), A075268 (trajectory of 442 in base 2), A077076 (trajectory of 537 in base 2), A077077 (trajectory of 775 in base 2), A066059 (trajectory of n in base 2 (presumably) does not reach a palindrome), A075252 (trajectory of n in base 2 does not reach a palindrome and (presumably) does not join the trajectory of any term m < n), A075153 (trajectory of 318 in base 4).
%Y A061561 Cf. A171470 (a(4*n)/2), A171471 (a(4*n+1)), A171472 (a(4*n+2)/12), A171473 (a(4*n+3)/3).
%K A061561 nonn,base
%O A061561 0,1
%A A061561 _N. J. A. Sloane_, May 18 2001
%E A061561 More terms from _Klaus Brockhaus_, May 27 2001