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A058042 Trajectory of binary number 10110 under the operation 'Reverse and Add!' carried out in base 2.

%I A058042 #40 Jan 31 2023 21:57:54
%S A058042 10110,100011,1010100,1101001,10110100,11100001,101101000,110010101,
%T A058042 1011101000,1101000101,10111010000,11000101101,101111010000,
%U A058042 110010001101,1011110100000,1100001011101,10111110100000
%N A058042 Trajectory of binary number 10110 under the operation 'Reverse and Add!' carried out in base 2.
%C A058042 According to J. Walker, Ronald Sprague has proved that this trajectory does not contain a palindrome. [I would like a reference for this.] Another proof has been given by _Klaus Brockhaus_.
%C A058042 10110 is the smallest number with this property in base 2. The analogous number in base 10 is believed to be 196, but its trajectory (see A006960) has never been proved not to contain a palindrome.
%C A058042 The binary numbers have a regular pattern with cycle length 4:
%C A058042   a(4k) = 101^(k+1)010^(k+1) for k >= 1,
%C A058042   a(4k+1) = 1101^(k-1)0001^(k-1)01 for k >= 2,
%C A058042   a(4k+2) = 101^(k+1)010^(k+2) for k >= 0,
%C A058042   a(4k+3) = 110^(k+1)101^(k)01 for k >= 1, where ^ stands for repeated concatenation. - _A.H.M. Smeets_, Feb 03 2019
%C A058042 From _A.H.M. Smeets_, Feb 11 2019: (Start)
%C A058042 Pattern with cycle length 4 represented by contextfree grammars with production rules:
%C A058042 S_a -> 10 T_a 00, T_a -> 1 T_a 0 | 1101;
%C A058042 S_b -> 11 T_b 01, T_b -> 0 T_b 1 | 1000;
%C A058042 S_c -> 10 T_c 000, T_c -> 1 T_c 0 | 1101;
%C A058042 S_d -> 11 T_d 101, T_d -> 0 T_d 1 | 0010;
%C A058042 see also A058042 for similar grammars for the binary represented trajectory of 77. (End)
%H A058042 T. D. Noe, <a href="/A058042/b058042.txt">Table of n, a(n) for n = 0..500</a>
%H A058042 T. Irvin, <a href="https://www.fourmilab.ch/documents/threeyears/two_months_more.html">About Two Months of Computing, or An Addendum to Mr. Walker's Three Years of Computing</a>, Aug 22 1995.
%H A058042 Klaus Brockhaus, <a href="/A058042/a058042.txt">On the 'Reverse and Add!' algorithm in base 2</a>
%H A058042 David J. Seal, <a href="https://www.mathpages.com/home/dseal.htm">Proofs similar to base 2 for base 4, 11, 17 and 26</a>
%H A058042 J. Walker, <a href="http://www.fourmilab.ch/documents/threeyears/threeyears.html">Three Years Of Computing: Final Report On The Palindrome Quest</a>
%H A058042 <a href="/index/Res#RAA">Index entries for sequences related to Reverse and Add!</a>
%F A058042 a(n) = A007088(A061561(n)). - _Reinhard Zumkeller_, Apr 21 2013
%t A058042 Clear[a]; a[0] = 10110; a[n_] := a[n] = (m = IntegerDigits[ a[n-1] ]; m2 = FromDigits[m, 2]; IntegerDigits[ FromDigits[m // Reverse, 2] + m2, 2] // FromDigits); Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Apr 03 2013 *)
%o A058042 (ARIBAS) var m,c,rev: integer; end; m := 22; c := 1; bit_write(m); write(" "); rev := bit_reverse(m); while m <> rev and c < 25 do inc(c); m := m + rev; bit_write(m); write(" "); rev := bit_reverse(m); end;
%o A058042 (Haskell)
%o A058042 a058042 = a007088 . a061561  -- _Reinhard Zumkeller_, Apr 21 2013
%Y A058042 See A061561 for the terms of A058042 written in base 10. Cf. A016016, A006960, A023108.
%K A058042 nonn,nice,base
%O A058042 0,1
%A A058042 _N. J. A. Sloane_, May 18 2001
%E A058042 More terms from _Klaus Brockhaus_, May 27 2001