

A334769


Numbers m that generate rotationally symmetrical XORtriangles T(m) that have central triangles of zeros.


12



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
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OFFSET

1,1


COMMENTS

An XORtriangle T(m) is an inverted 01 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 zerotriangle (CZT) is a field of contiguous 0bits 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".


LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Central zerotriangles in rotationally symmetrical XORTriangles, 2020.
Michael De Vlieger, Basic aspects of rotationally symmetrical XORtriangles that have central zero triangles
Michael De Vlieger, Diagram montage of XORtriangles 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 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. 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 XORtriangle has many bubbles but none in the center, so 91 is not in the sequence.


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] ]


PROG

(C) See Links section.


CROSSREFS

Cf. A038554, A070939, A334556, A334770, A334771, A334796, A334836.
Sequence in context: A247346 A300394 A142225 * A334931 A059858 A152310
Adjacent sequences: A334766 A334767 A334768 * A334770 A334771 A334772


KEYWORD

nonn


AUTHOR

Michael De Vlieger, May 10 2020


EXTENSIONS

Data corrected by Rémy Sigrist, May 15 2020


STATUS

approved



