A334770 Side length s of the central triangle of zeros in the XOR-triangle T(n).

2, 2, 1, 4, 4, 1, 2, 2, 2, 2, 3, 6, 6, 3, 5, 8, 2, 2, 2, 2, 8, 5, 6, 3, 3, 6, 6, 3, 3, 6, 1, 1, 1, 1, 7, 10, 4, 4, 4, 4, 10, 7, 1, 1, 1, 1, 3, 9, 3, 12, 3, 6, 3, 6, 6, 3, 6, 3, 12, 3, 9, 3, 1, 1, 1, 1, 10, 7, 4, 4, 7, 10, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 10

Offset: 1

Author

Michael De Vlieger, May 10 2020

Keywords

• nonn

Comments

An XOR-triangle T(n) is an inverted 0-1 triangle formed by choosing a top row the binary rendition of n and having each entry in subsequent rows be the XOR of the two values above it, i.e., A038554(n) applied recursively until we reach a single bit.

A334556 is the sequence of rotationally symmetrical T(n).

A central zero-triangle (CZT) is a field of contiguous 0-bits in T(n) surrounded on all sides by a layer of 1 bits, and generally k > 1 bits of any parity. Alternatively, these might be referred to as "central bubbles".

Examples

			For n = 151, we have rotationally symmetrical T(151) as below, replacing 0 with "." for clarity:
  1 . . 1 . 1 1 1
   1 . 1 1 1 . .
    1 1 . . 1 .
     . 1 . 1 1
      1 1 1 .
       . . 1
        . 1
         1
At the center of the figure we see a CZT with s = 2, ringed by 1s, with k = 2. Since 151 is the first term of A334769, a(1) = 2.
For n = 599, we have a rotationally symmetrical T(599) with s = 4 and k = 2.
  1 . . 1 . 1 . 1 1 1
   1 . 1 1 1 1 1 . .
    1 1 . . . . 1 .
     . 1 . . . 1 1
      1 1 . . 1 .
       . 1 . 1 1
        1 1 1 .
         . . 1
          . 1
           1
Since A334769(4) = 599, a(4) = 4.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Central zero-triangles in rotationally symmetrical XOR-Triangles, 2020.
• Index entries for sequences related to binary expansion of n
• Index entries for sequences related to XOR-triangles

Crossrefs

Cf. A038554, A070939, A334556, A334769, A334771, A334796.

Programs

Mathematica

Block[{f, s = Rest[Import["https://oeis.org/A334556/b334556.txt", "Data"][[All, -1]] ]}, f[n_] := NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[n, 2], Length@ # > 1 &]; Array[Block[{n = s[[#]]}, If[# == 0, Nothing, #] &@ FirstCase[MapIndexed[If[2 #2 > #3 + 1, Nothing, #1[[#2 ;; -#2]]] & @@ {#1, First[#2], Length@ #1} &, f[n][[1 ;; Ceiling[IntegerLength[#, 2]/(2 Sqrt[3])] + 3]] ], r_List /; FreeQ[r, 1] :> Length@ r] /. k_ /; MissingQ@ k -> 0] &, Lengths - 1, 2] ]