|
%I #40 Jan 31 2023 21:57:54
%S 10110,100011,1010100,1101001,10110100,11100001,101101000,110010101,
%T 1011101000,1101000101,10111010000,11000101101,101111010000,
%U 110010001101,1011110100000,1100001011101,10111110100000
%N Trajectory of binary number 10110 under the operation 'Reverse and Add!' carried out in base 2.
%C 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 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 The binary numbers have a regular pattern with cycle length 4:
%C a(4k) = 101^(k+1)010^(k+1) for k >= 1,
%C a(4k+1) = 1101^(k-1)0001^(k-1)01 for k >= 2,
%C a(4k+2) = 101^(k+1)010^(k+2) for k >= 0,
%C 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 From _A.H.M. Smeets_, Feb 11 2019: (Start)
%C Pattern with cycle length 4 represented by contextfree grammars with production rules:
%C S_a -> 10 T_a 00, T_a -> 1 T_a 0 | 1101;
%C S_b -> 11 T_b 01, T_b -> 0 T_b 1 | 1000;
%C S_c -> 10 T_c 000, T_c -> 1 T_c 0 | 1101;
%C S_d -> 11 T_d 101, T_d -> 0 T_d 1 | 0010;
%C see also A058042 for similar grammars for the binary represented trajectory of 77. (End)
%H T. D. Noe, <a href="/A058042/b058042.txt">Table of n, a(n) for n = 0..500</a>
%H 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 Klaus Brockhaus, <a href="/A058042/a058042.txt">On the 'Reverse and Add!' algorithm in base 2</a>
%H 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 J. Walker, <a href="http://www.fourmilab.ch/documents/threeyears/threeyears.html">Three Years Of Computing: Final Report On The Palindrome Quest</a>
%H <a href="/index/Res#RAA">Index entries for sequences related to Reverse and Add!</a>
%F a(n) = A007088(A061561(n)). - _Reinhard Zumkeller_, Apr 21 2013
%t 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 (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 (Haskell)
%o a058042 = a007088 . a061561 -- _Reinhard Zumkeller_, Apr 21 2013
%Y See A061561 for the terms of A058042 written in base 10. Cf. A016016, A006960, A023108.
%K nonn,nice,base
%O 0,1
%A _N. J. A. Sloane_, May 18 2001
%E More terms from _Klaus Brockhaus_, May 27 2001
| 