A334769 Numbers m that generate rotationally symmetrical XOR-triangles T(m) that have central triangles of zeros.

151, 233, 543, 599, 937, 993, 1379, 1483, 1589, 1693, 2359, 2391, 3753, 3785, 8607, 9559, 10707, 11547, 13029, 13869, 15017, 15969, 22115, 22243, 23627, 23755, 25397, 25525, 26909, 27037, 33151, 34591, 35535, 36015, 37687, 38231, 39047, 40679, 57625, 59257

Offset

1,1

Comments

An XOR-triangle T(m) is an inverted 0-1 triangle formed by choosing as 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(m) (here abbreviated RST).

A central zero-triangle (CZT) is a field of contiguous 0-bits in T(m) 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

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Central zero-triangles in rotationally symmetrical XOR-Triangles, 2020.

Michael De Vlieger, Basic aspects of rotationally symmetrical XOR-triangles that have central zero triangles

Michael De Vlieger, Diagram montage of XOR-triangles for terms 1 <= n <= 1000.

Rémy Sigrist, C program for A334769K

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. Thus 151 is in the sequence.
For n = 11, we have rotationally symmetrical T(11):
  1 . 1 1
   1 1 .
    . 1
     1
Since there is no CZT, 11 is not in the sequence.
For n = 91, we have rotationally symmetrical T(91):
  1 . 1 1 . 1 1
   1 1 . 1 1 .
    . 1 1 . 1
     1 . 1 1
      1 1 .
       . 1
        1
This XOR-triangle has many bubbles but none in the center, so 91 is not in the sequence.

Mathematica

Block[{s, t = Array[NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[#, 2], Length@ # > 1 &] &, 2^18]}, s = Select[Range[Length@ t], Function[{n, h}, (Reverse /@ Transpose[MapIndexed[PadRight[#, h, -1] &, t[[n]] ]] /. -1 -> Nothing) == t[[n]]] @@ {#, IntegerLength[#, 2]} &]; Array[Block[{n = s[[#]]}, If[# == 0, Nothing, n] &@ FirstCase[MapIndexed[If[2 #2 > #3 + 1, Nothing, #1[[#2 ;; -#2]]] & @@ {#1, First[#2], Length@ #1} &, NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[n, 2], Length@ # > 1 &][[1 ;; Ceiling[IntegerLength[#, 2]/(2 Sqrt[3])] + 3]]  ], r_List /; FreeQ[r, 1] :> Length@ r] /. k_ /; MissingQ@ k -> 0] &, Length@ s - 1, 2] ]

Prog

(C) See Links section.

Crossrefs

Cf. A038554, A070939, A334556, A334770, A334771, A334796, A334836.

Sequence in context: A247346 A300394 A142225 * A334931 A059858 A152310

Adjacent sequences: A334766 A334767 A334768 * A334770 A334771 A334772

Keyword

nonn

Author

Michael De Vlieger, May 10 2020

Extensions

Data corrected by Rémy Sigrist, May 15 2020

Status

approved