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

Author

Michael De Vlieger, May 10 2020

Keywords

• nonn

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".

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. 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.
		

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

Crossrefs

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

Programs

C

See Links section.

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] ]

Extensions

Data corrected by Rémy Sigrist, May 15 2020