OFFSET
0,1
COMMENTS
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.
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.
The binary numbers have a regular pattern with cycle length 4:
a(4k) = 101^(k+1)010^(k+1) for k >= 1,
a(4k+1) = 1101^(k-1)0001^(k-1)01 for k >= 2,
a(4k+2) = 101^(k+1)010^(k+2) for k >= 0,
a(4k+3) = 110^(k+1)101^(k)01 for k >= 1, where ^ stands for repeated concatenation. - A.H.M. Smeets, Feb 03 2019
From A.H.M. Smeets, Feb 11 2019: (Start)
Pattern with cycle length 4 represented by contextfree grammars with production rules:
S_a -> 10 T_a 00, T_a -> 1 T_a 0 | 1101;
S_b -> 11 T_b 01, T_b -> 0 T_b 1 | 1000;
S_c -> 10 T_c 000, T_c -> 1 T_c 0 | 1101;
S_d -> 11 T_d 101, T_d -> 0 T_d 1 | 0010;
see also A058042 for similar grammars for the binary represented trajectory of 77. (End)
LINKS
T. D. Noe, Table of n, a(n) for n = 0..500
T. Irvin, About Two Months of Computing, or An Addendum to Mr. Walker's Three Years of Computing, Aug 22 1995.
Klaus Brockhaus, On the 'Reverse and Add!' algorithm in base 2
David J. Seal, Proofs similar to base 2 for base 4, 11, 17 and 26
FORMULA
MATHEMATICA
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 *)
PROG
(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;
(Haskell)
a058042 = a007088 . a061561 -- Reinhard Zumkeller, Apr 21 2013
CROSSREFS
KEYWORD
nonn,nice,base
AUTHOR
N. J. A. Sloane, May 18 2001
EXTENSIONS
More terms from Klaus Brockhaus, May 27 2001
STATUS
approved