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A334931 Numbers that generate rotationally symmetrical XOR-triangles with a pattern of zero-triangles of edge length 2, some of which are clipped to result in some singleton zeros at the edges. 3
151, 233, 1483, 1693, 10707, 13029, 644007, 941241, 317049751, 490370281, 3111314891, 3550957213, 22455577043, 27325461221, 1350581212071, 1973926386873, 664901519788951, 1028381017273577, 6524900247528907, 7446897021636253, 47092758308252115, 57305645652210405 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Subset of A334769 which is a subset of A334556.

Numbers m in this sequence A070939(m) (mod 3) = 2. The numbers in this sequence can be constructed using run lengths of bits.

2n has the reverse run length pattern as 2n - 1. a(1) has the run lengths {1, 2, 1, 1, 3}, while a(2) has {3, 1, 1, 2, 1}, etc.

For n = 1 (mod 8): 12..(1132)..113;

For n = 3 (mod 8): 113..(2113)..2112;

For n = 5 (mod 8): 11123..(1123)..1122;

For n = 7 (mod 8): 123112..(3112)..31123, where the parenthetic run lengths occur, when they occur, in multiples of 3. Thus, a(9) has the run length form 12113211321132113 = binary 10010111001011100101110010111 = decimal 317049751.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..1264

Michael De Vlieger, Diagram montage of XOR-triangles resulting from a(n) with 1 <= n <= 32.

Michael De Vlieger, Central zero-triangles in rotationally symmetrical XOR-Triangles, 2020.

Index entries for sequences related to binary expansion of n

Index entries for sequences related to XOR-triangles

EXAMPLE

Diagrams of a(1)-a(6), replacing "0" with "." and "1" with "@" for clarity:

a(1) = 151 (a(2) = 233 appears as a mirror image):

  @ . . @ . @ @ @

   @ . @ @ @ . .

    @ @ . . @ .

     . @ . @ @

      @ @ @ .

       . . @

        . @

         @

.

a(3) = 1483 (a(4) = 1693 appears as a mirror image):

  @ . @ @ @ . . @ . @ @

   @ @ . . @ . @ @ @ .

    . @ . @ @ @ . . @

     @ @ @ . . @ . @

      . . @ . @ @ @

       . @ @ @ . .

        @ . . @ .

         @ . @ @

          @ @ .

           . @

            @

.

a(5) = 10707 (a(6) = 13029 appears as a mirror image):

  @ . @ . . @ @ @ . @ . . @ @

   @ @ @ . @ . . @ @ @ . @ .

    . . @ @ @ . @ . . @ @ @

     . @ . . @ @ @ . @ . .

      @ @ . @ . . @ @ @ .

       . @ @ @ . @ . . @

        @ . . @ @ @ . @

         @ . @ . . @ @

          @ @ @ . @ .

           . . @ @ @

            . @ . .

             @ @ .

              . @

               @

MATHEMATICA

Array[FromDigits[Flatten@ MapIndexed[ConstantArray[#2, #1] & @@ {#1, Mod[First[#2], 2]} &, If[EvenQ@ #1, Reverse@ #2, #2]], 2] & @@ {#1, Which[#2 == 1, PadRight[{1, 2}, 12 Ceiling[#1/8] - 7, {3, 2, 1, 1}], #2 == 2, PadRight[{1, 1}, 12 Ceiling[#1/8] - 6, {1, 1, 3, 2}]~Join~{2}, #2 == 3, PadRight[{1, 1}, 12 Ceiling[#1/8] - 4, {3, 1, 1, 2}]~Join~{2}, True, PadRight[{}, 12 Ceiling[#1/8] - 1, {1, 2, 3, 1}]]} & @@ {#, Ceiling[Mod[#, 8]/2]} &, 22]

(* Generate a textual plot of XOR-triangle T(n) *)

xortri[n_Integer] := TableForm@ MapIndexed[StringJoin[ConstantArray[" ", First@ #2 - 1], StringJoin @@ Riffle[Map[If[# == 0, "." (* 0 *), "@" (* 1 *)] &, #1], " "]] &, NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[n, 2], Length@ # > 1 &]]

CROSSREFS

Cf. A334556, A334769, A334930, A334932.

Sequence in context: A300394 A142225 A334769 * A059858 A152310 A276264

Adjacent sequences:  A334927 A334928 A334930 * A334932 A334933 A334934

KEYWORD

nonn,easy

AUTHOR

Michael De Vlieger, May 16 2020

STATUS

approved

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Last modified August 9 13:39 EDT 2020. Contains 336323 sequences. (Running on oeis4.)