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A334932 Numbers that generate rotationally symmetrical XOR-triangles with a pattern of zero-triangles of edge length 3, some of which are clipped to result in some zero-triangles of edge length 2 at the edges. 3
2535, 3705, 162279, 237177, 10385895, 15179385, 664697319, 971480697, 42540628455, 62174764665, 2722600221159, 3979184938617, 174246414154215, 254667836071545, 11151770505869799, 16298741508578937, 713713312375667175, 1043119456549052025, 45677651992042699239 (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) = 0. All m have first and last bits = 1.

The numbers in this sequence can be constructed using run lengths of bits thus: 12..(42)..3 or the reverse 3..(24)..21, with at least one copy of the pair of parenthetic numbers.

Thus, the smallest number m has run lengths {1, 2, 4, 2, 3}, which is the binary 100111100111 = decimal 2535.

2n has the reverse run length pattern as 2n - 1. a(3) has the run lengths {1, 2, 4, 2, 4, 2, 3}, while a(4) has {3, 2, 4, 2, 4, 2, 1}, etc.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..1104

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 linear recurrences with constant coefficients, signature (0,65,0,-64).

Index entries for sequences related to XOR-triangles

FORMULA

From Colin Barker, Jun 09 2020: (Start)

G.f.: 3*x*(13 + 19*x)*(65 - 64*x^2) / ((1 - x)*(1 + x)*(1 - 8*x)*(1 + 8*x)).

a(n) = 65*a(n-2) - 64*a(n-4) for n>4.

a(n) = (1/21)*(-16 - 3*(-1)^n + 123*2^(5+3*n) - 85*(-1)^n*2^(5 + 3*n)) for n>0.

(End)

EXAMPLE

Diagrams of a(1)-a(4), replacing “0” with “.” and “1” with “@” for clarity:

a(1) = 2535 (a(2) = 3705 appears as a mirror image):

  @ . . @ @ @ @ . . @ @ @

   @ . @ . . . @ . @ . .

    @ @ @ . . @ @ @ @ .

     . . @ . @ . . . @

      . @ @ @ @ . . @

       @ . . . @ . @

        @ . . @ @ @

         @ . @ . .

          @ @ @ .

           . . @

            . @

             @

.

a(3) = 162279 (a(4) = 237177 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] & @@ {#, Join[{1, 2}, PadRight[{}, Ceiling[#, 2], {4, 2}], {3}]} &, 19]

(* 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, A334931.

Sequence in context: A231968 A172142 A206596 * A254021 A254014 A254888

Adjacent sequences:  A334929 A334930 A334931 * A334933 A334934 A334935

KEYWORD

nonn,easy

AUTHOR

Michael De Vlieger, May 16 2020

STATUS

approved

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Last modified November 29 14:50 EST 2021. Contains 349416 sequences. (Running on oeis4.)