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

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, 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, 1, 2, 1, 4, 1, 4, 1, 6, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3

Offset

1,5

Comments

See A367508 for the description of the Christmas tree patterns, references and links.

Links

Paolo Xausa, Table of n, a(n) for n = 1..13494 (first 15 orders).

Example

The first 4 tree pattern orders are shown below (left), with the corresponding number of even terms on the right.
.
Order 1:                        |
              0  1              |  1
                                |
Order 2:                        |
               10               |  1
           00  01  11           |  1
                                |
Order 3:                        |
            100  101            |  1
            010  110            |  2
       000  001  011  111       |  1
                                |
Order 4:                        |
              1010              |  1
        1000  1001  1011        |  1
              1100              |  1
        0100  0101  1101        |  1
        0010  0110  1110        |  3
  0000  0001  0011  0111  1111  |  1
.

Mathematica

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 *)

Prog

(Python)
from itertools import islice
from functools import reduce
def uniq(r): return reduce(lambda u, e: u if e in u else u+[e], r, [])
def agen():  # generator of terms
    R = [["0", "1"]]
    while R:
        r = R.pop(0)
        yield sum(b[-1] == '0' for b in r)
        if len(r) > 1: R.append(uniq([r[k]+"0" for k in range(1, len(r))]))
        R.append(uniq([r[0]+"0", r[0]+"1"] + [r[k]+"1" for k in range(1, len(r))]))
print(list(islice(agen(), 88))) # Michael S. Branicky, Dec 25 2023

Crossrefs

Cf. A367508, A368463, A368464.

Sequence in context: A333769 A259396 A328672 * A378222 A325355 A219093

Adjacent sequences: A368462 A368463 A368464 * A368466 A368467 A368468

Keyword

nonn

Author

Paolo Xausa, Dec 25 2023

Status

approved