%I #44 Aug 04 2024 19:02:54
%S 1,1,2,1,2,2,2,1,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,
%T 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,
%U 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2
%N Number of distinct base-2 digits of n.
%C Also, if prefixed by 0, the trajectory of 0 under repeated applications of the morphism 0 -> 0,1, 1 -> 1,2, 2 -> 2,2. This is a word that is pure uniform morphic, but neither primitive morphic nor recurrent. - _N. J. A. Sloane_, Jul 15 2018
%D Dekking, Michel, Michel Mendes France, and Alf van der Poorten. "Folds." The Mathematical Intelligencer, 4.3 (1982): 130-138 & front cover, and 4:4 (1982): 173-181 (printed in two parts). See Observaion 1.8.
%H Antti Karttunen, <a href="/A043529/b043529.txt">Table of n, a(n) for n = 0..16384</a>
%H Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, Luca Q. Zamboni, <a href="https://arxiv.org/abs/1711.10807">A Taxonomy of Morphic Sequences</a>, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017. See Example 35.
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F This is 2 unless n = 2^k - 1 for some k in which case it is 1.
%F a(n) = 2 - A036987(n). - _Antti Karttunen_, Nov 19 2017
%p A043529 := proc(n): if type(ln(n+1)/ln(2), integer) then 1 else 2 fi: end proc: seq(A043529(n), n=0..90); # _Johannes W. Meijer_, Sep 14 2012
%t (* Needs version >= 10.2. *)
%t SubstitutionSystem[{0 -> {0, 1}, 1 -> {1, 2}, 2 -> {2, 2}}, 0, 7] // Last // Rest (* _Jean-François Alcover_, Apr 06 2020 *)
%t Table[Length[Union[IntegerDigits[n,2]]],{n,0,90}] (* _Harvey P. Dale_, Aug 04 2024 *)
%Y Factor of A160466. Cf. A007456 and A081729. - _Johannes W. Meijer_, May 24 2009
%Y Cf. A007088, A036987, A043545.
%Y Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.
%K nonn,base,easy
%O 0,3
%A _Clark Kimberling_
%E First term added and offset changed by _Johannes W. Meijer_, May 15 2009