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A368464 Number of odd terms in each row of the iterates of the Christmas tree pattern map (A367508).

%I A368464 #17 Dec 29 2023 13:54:46
%S A368464 1,0,2,1,0,3,0,2,0,2,0,4,1,0,3,1,0,3,0,3,0,5,0,2,0,2,0,4,0,2,0,2,0,4,
%T A368464 0,2,0,4,0,4,0,6,1,0,3,1,0,3,0,3,0,5,1,0,3,1,0,3,0,3,0,5,1,0,3,0,3,0,
%U A368464 5,0,3,0,5,0,5,0,7,0,2,0,2,0,4,0,2,0,2,0
%N A368464 Number of odd terms in each row of the iterates of the Christmas tree pattern map (A367508).
%C A368464 See A367508 for the description of the Christmas tree patterns, references and links.
%H A368464 Paolo Xausa, <a href="/A368464/b368464.txt">Table of n, a(n) for n = 1..13494</a> (first 15 orders).
%e A368464 The first 4 tree pattern orders are shown below (left), with the corresponding number of odd terms on the right.
%e A368464 .
%e A368464 Order 1:                        |
%e A368464               0  1              |  1
%e A368464                                 |
%e A368464 Order 2:                        |
%e A368464                10               |  0
%e A368464            00  01  11           |  2
%e A368464                                 |
%e A368464 Order 3:                        |
%e A368464             100  101            |  1
%e A368464             010  110            |  0
%e A368464        000  001  011  111       |  3
%e A368464                                 |
%e A368464 Order 4:                        |
%e A368464               1010              |  0
%e A368464         1000  1001  1011        |  2
%e A368464               1100              |  0
%e A368464         0100  0101  1101        |  2
%e A368464         0010  0110  1110        |  0
%e A368464   0000  0001  0011  0111  1111  |  4
%e A368464 .
%e A368464 Apparently, removing the 0 terms from the order i pattern (for i >= 2), gives the row lengths of the order i-1 pattern (cf. A363718).
%t A368464 With[{imax=8},Map[Total,Map[Mod[FromDigits[#],2]&,NestList[Map[Delete[{If[Length[#]>1,Map[#<>"0"&,Rest[#]],Nothing],Join[{#[[1]]<>"0"},Map[#<>"1"&,#]]},0]&],{{"0","1"}},imax-1],{3}],{2}]] (* Generates terms up to order 8 *)
%o A368464 (Python)
%o A368464 from itertools import islice
%o A368464 from functools import reduce
%o A368464 def uniq(r): return reduce(lambda u, e: u if e in u else u+[e], r, [])
%o A368464 def agen():  # generator of terms
%o A368464     R = [["0", "1"]]
%o A368464     while R:
%o A368464         r = R.pop(0)
%o A368464         yield sum(1 for b in r if b[-1] == '1')
%o A368464         if len(r) > 1: R.append(uniq([r[k]+"0" for k in range(1, len(r))]))
%o A368464         R.append(uniq([r[0]+"0", r[0]+"1"] + [r[k]+"1" for k in range(1, len(r))]))
%o A368464 print(list(islice(agen(), 88))) # _Michael S. Branicky_, Dec 25 2023
%Y A368464 Cf. A363718, A367508, A368463, A368465.
%K A368464 nonn
%O A368464 1,3
%A A368464 _Paolo Xausa_, Dec 25 2023