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A368465 Number of even terms in each row of the iterates of the Christmas tree pattern map (A367508).

%I A368465 #14 Dec 29 2023 13:52:48
%S A368465 1,1,1,1,2,1,1,1,1,1,3,1,1,2,1,1,2,1,2,1,4,1,1,1,1,1,3,1,1,1,1,1,3,1,
%T A368465 1,1,3,1,3,1,5,1,1,2,1,1,2,1,2,1,4,1,1,2,1,1,2,1,2,1,4,1,1,2,1,2,1,4,
%U A368465 1,2,1,4,1,4,1,6,1,1,1,1,1,3,1,1,1,1,1,3
%N A368465 Number of even terms in each row of the iterates of the Christmas tree pattern map (A367508).
%C A368465 See A367508 for the description of the Christmas tree patterns, references and links.
%H A368465 Paolo Xausa, <a href="/A368465/b368465.txt">Table of n, a(n) for n = 1..13494</a> (first 15 orders).
%e A368465 The first 4 tree pattern orders are shown below (left), with the corresponding number of even terms on the right.
%e A368465 .
%e A368465 Order 1:                        |
%e A368465               0  1              |  1
%e A368465                                 |
%e A368465 Order 2:                        |
%e A368465                10               |  1
%e A368465            00  01  11           |  1
%e A368465                                 |
%e A368465 Order 3:                        |
%e A368465             100  101            |  1
%e A368465             010  110            |  2
%e A368465        000  001  011  111       |  1
%e A368465                                 |
%e A368465 Order 4:                        |
%e A368465               1010              |  1
%e A368465         1000  1001  1011        |  1
%e A368465               1100              |  1
%e A368465         0100  0101  1101        |  1
%e A368465         0010  0110  1110        |  3
%e A368465   0000  0001  0011  0111  1111  |  1
%e A368465 .
%t A368465 With[{imax=8},Map[Total,Map[Mod[FromDigits[#]+1,2]&,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 A368465 (Python)
%o A368465 from itertools import islice
%o A368465 from functools import reduce
%o A368465 def uniq(r): return reduce(lambda u, e: u if e in u else u+[e], r, [])
%o A368465 def agen():  # generator of terms
%o A368465     R = [["0", "1"]]
%o A368465     while R:
%o A368465         r = R.pop(0)
%o A368465         yield sum(b[-1] == '0' for b in r)
%o A368465         if len(r) > 1: R.append(uniq([r[k]+"0" for k in range(1, len(r))]))
%o A368465         R.append(uniq([r[0]+"0", r[0]+"1"] + [r[k]+"1" for k in range(1, len(r))]))
%o A368465 print(list(islice(agen(), 88))) # _Michael S. Branicky_, Dec 25 2023
%Y A368465 Cf. A367508, A368463, A368464.
%K A368465 nonn
%O A368465 1,5
%A A368465 _Paolo Xausa_, Dec 25 2023