OFFSET
1,1
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
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.
Michael De Vlieger, Diagram montage showing the first dozen XOR-triangles exhibiting frame widths of 2, 3, 4, ..., 12 by row.
EXAMPLE
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)
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] ]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, May 12 2020
STATUS
approved