A334796 a(n) = (A070939(A334769(n)) - A334770(n))/3.

2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 2, 4, 4, 4, 4, 2, 3, 3, 4, 4, 3, 3, 4, 4, 3, 5, 5, 5, 5, 3, 2, 4, 4, 4, 4, 2, 3, 5, 5, 5, 5, 5, 3, 5, 2, 5, 4, 5, 4, 4, 5, 4, 5, 2, 5, 3, 5, 6, 6, 6, 6, 3, 4, 5, 5, 4, 3, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 3, 4, 5

Offset: 1

Author

Michael De Vlieger, May 12 2020

Keywords

• nonn

Comments

An XOR-triangle T(m) 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(m) applied recursively until we reach a single bit.

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

A central zero-triangle (CZT) is a field of contiguous 0-bits, listed in A334769, a subset of A334556. CZTs have side length k = A334770(n), surrounded on all sides by a layer of 1 bits, and generally j > 1 bits of any parity.

This sequence describes the "frame width" j.

Smallest n with a given value of j appears in A334836. - Michael De Vlieger, May 20 2020

Examples

			a(4) pertains to T(599), with A334770(4) = 4.
(1 + A070939(599) - 4)/3 = (1 + 9 - 4)/3 = 6/3 = 2, thus a(4) = 2.
(Diagram, replacing 0 with “.”):
  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
a(11) pertains to T(2359), with A334770(11) = 3.
(1 + A070939(2359) - 4)/3 = (1 + 11 - 3)/3 = 9/3 = 3, thus a(11) = 3.
(Diagram):
  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 1 . 1 1 .
        . . 1 1 . 1
         . 1 . 1 1
          1 1 1 .
           . . 1
            . 1
             1
From _Michael De Vlieger_, May 14 2020: (Start)
Linear recurrences that produce XOR-triangles with frame length j (table may not be exhaustive):
j   LR          Lower               Upper
-----------------------------------------------------
2   (5, -4)     {39, 151}           {57, 223}
3   (17, -16)   {543, 8607}         {993, 15969}
                {1379, 22115}       {1589, 25397}
                {1483, 23755}       {1693, 27037}
                {2359, 37687}       {3785, 60617}
4   (17, -16)   {22243, 356067}     {25525, 408501}
                {39047, 624775}     {57625, 921881}
                {40679, 650983}     {59257, 948089}
                {171475, 2743763}   {208613, 3337957}
                {356067, 5697251}   {408501, 6536117}
... (End)
		

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.
• Michael De Vlieger, Diagram montage showing XOR-triangles for terms in certain linear recurrences and their bit-reversals, illustrating relations in their appearance, most significantly, constant frame width.
• Index entries for sequences related to binary expansion of n
• Index entries for sequences related to XOR-triangles
• Michael De Vlieger, Diagram montage showing the first dozen XOR-triangles exhibiting frame widths of 2, 3, 4, ..., 12 by row.

Crossrefs

Cf. A038554, A070939, A334556, A334769, A334770, A334836.

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, (1 + Floor@ Log2[n] - #)/3] &@ 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] &, Length@ s - 1, 2] ]