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A334769 Numbers m that generate rotationally symmetrical XOR-triangles T(m) that have central triangles of zeros. 12

%I

%S 151,233,543,599,937,993,1379,1483,1589,1693,2359,2391,3753,3785,8607,

%T 9559,10707,11547,13029,13869,15017,15969,22115,22243,23627,23755,

%U 25397,25525,26909,27037,33151,34591,35535,36015,37687,38231,39047,40679,57625,59257

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

%C 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.

%C A334556 is the sequence of rotationally symmetrical T(m) (here abbreviated RST).

%C 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".

%H Rémy Sigrist, <a href="/A334769/b334769.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="http://vincico.com/seq/a334769.html">Central zero-triangles in rotationally symmetrical XOR-Triangles</a>, 2020.

%H Michael De Vlieger, <a href="/A334769/a334769_2.txt">Basic aspects of rotationally symmetrical XOR-triangles that have central zero triangles</a>

%H Michael De Vlieger, <a href="/A334769/a334769_3.png">Diagram montage</a> of XOR-triangles for terms 1 <= n <= 1000.

%H Rémy Sigrist, <a href="/A334769/a334769_1.txt">C program for A334769K</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%H <a href="/index/X#XOR-triangles">Index entries for sequences related to XOR-triangles</a>

%e For n = 151, we have rotationally symmetrical T(151) as below, replacing 0 with "." for clarity:

%e 1 . . 1 . 1 1 1

%e 1 . 1 1 1 . .

%e 1 1 . . 1 .

%e . 1 . 1 1

%e 1 1 1 .

%e . . 1

%e . 1

%e 1

%e 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.

%e For n = 11, we have rotationally symmetrical T(11):

%e 1 . 1 1

%e 1 1 .

%e . 1

%e 1

%e Since there is no CZT, 11 is not in the sequence.

%e For n = 91, we have rotationally symmetrical T(91):

%e 1 . 1 1 . 1 1

%e 1 1 . 1 1 .

%e . 1 1 . 1

%e 1 . 1 1

%e 1 1 .

%e . 1

%e 1

%e This XOR-triangle has many bubbles but none in the center, so 91 is not in the sequence.

%t 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] ]

%o (C) See Links section.

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

%K nonn

%O 1,1

%A _Michael De Vlieger_, May 10 2020

%E Data corrected by _Rémy Sigrist_, May 15 2020

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Last modified September 23 16:17 EDT 2020. Contains 337314 sequences. (Running on oeis4.)