

A334770


Side length s of the central triangle of zeros in the XORtriangle 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
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OFFSET

1,1


COMMENTS

An XORtriangle T(n) is an inverted 01 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 zerotriangle (CZT) is a field of contiguous 0bits 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 zerotriangles in rotationally symmetrical XORTriangles, 2020.
Index entries for sequences related to binary expansion of n
Index entries for sequences related to XORtriangles


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



