login
Number of zeros (or ones) in each row of the iterates of the Christmas tree pattern map (A367508).
4

%I #21 Nov 29 2023 07:46:04

%S 1,1,3,3,3,6,2,6,2,6,6,10,5,5,10,5,5,10,5,10,10,15,3,9,3,9,9,15,3,9,3,

%T 9,9,15,3,9,9,15,9,15,15,21,7,7,14,7,7,14,7,14,14,21,7,7,14,7,7,14,7,

%U 14,14,21,7,7,14,7,14,14,21,7,14,14,21,14,21,21,28

%N Number of zeros (or ones) in each row of the iterates of the Christmas tree pattern map (A367508).

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

%H Paolo Xausa, <a href="/A367555/b367555.txt">Table of n, a(n) for n = 1..13494</a> (first 15 orders).

%e The following diagram shows the first 4 tree pattern orders, along with the corresponding number of zeros = number of ones.

%e .

%e Order 1: |

%e 0 1 | 1

%e |

%e Order 2: |

%e 10 | 1

%e 00 01 11 | 3

%e |

%e Order 3: |

%e 100 101 | 3

%e 010 110 | 3

%e 000 001 011 111 | 6

%e |

%e Order 4: |

%e 1010 | 2

%e 1000 1001 1011 | 6

%e 1100 | 2

%e 0100 0101 1101 | 6

%e 0010 0110 1110 | 6

%e 0000 0001 0011 0111 1111 | 10

%e .

%t With[{imax=9},Map[Total,NestList[Map[Delete[{If[Length[#]>1,Rest[#],Nothing],Join[{First[#]},#+1]},0]&],{{0,1}},imax-1],{2}]] (* Generates terms up to order 9 *)

%o (Python)

%o from itertools import islice

%o from functools import reduce

%o def uniq(r): return reduce(lambda u, e: u if e in u else u+[e], r, [])

%o def agen(): # generator of terms

%o R = [["0", "1"]]

%o while R:

%o r = R.pop(0)

%o yield sum(e.count("1") for e in r)

%o if len(r) > 1: R.append(uniq([r[k]+"0" for k in range(1, len(r))]))

%o R.append(uniq([r[0]+"0", r[0]+"1"] + [r[k]+"1" for k in range(1, len(r))]))

%o print(list(islice(agen(), 77))) # _Michael S. Branicky_, Nov 23 2023

%Y Cf. A367508, A367562.

%K nonn,base

%O 1,3

%A _Paolo Xausa_, Nov 22 2023