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A334770 Side length s of the central triangle of zeros in the XOR-triangle T(n). 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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".

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

EXAMPLE

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.

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] ]

CROSSREFS

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

Sequence in context: A276477 A099491 A322082 * A274622 A138189 A110090

Adjacent sequences:  A334767 A334768 A334769 * A334771 A334772 A334773

KEYWORD

nonn

AUTHOR

Michael De Vlieger, May 10 2020

STATUS

approved

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Last modified September 23 15:43 EDT 2020. Contains 337310 sequences. (Running on oeis4.)