A334931 -id:A334931 - cp's OEIS frontend

A334930 Numbers that generate rotationally symmetrical XOR-triangles featuring singleton zero bits in a hexagonal arrangement.

1, 11, 13, 91, 109, 731, 877, 5851, 7021, 46811, 56173, 374491, 449389, 2995931, 3595117, 23967451, 28760941, 191739611, 230087533, 1533916891, 1840700269, 12271335131, 14725602157, 98170681051, 117804817261, 785365448411, 942438538093, 6282923587291, 7539508304749

Offset: 1

Michael De Vlieger, May 16 2020

Subset of A334556.
No zero appears in the center of the figure, thus a(n) does not intersect A334769.
Numbers m with A070939(m) (mod 3) = 1 involving alternating run lengths of a singleton zero separated by a pair of 1s in the binary expansion, admitting an initial or final singleton 1.

			Diagrams of a(2)-a(5), replacing “0” with “.” and “1” with “@” for clarity:
     a(2)=11            a(3)=13
     @ . @ @            @ @ . @
      @ @ .              . @ @
       . @                @ .
        @                  @
.
    a(4) = 91          a(5) = 109
  @ . @ @ . @ @      @ @ . @ @ . @
   @ @ . @ @ .        . @ @ . @ @
    . @ @ . @          @ . @ @ .
     @ . @ @            @ @ . @
      @ @ .              . @ @
       . @                @ .
        @                  @

Michael De Vlieger, Table of n, a(n) for n = 1..2215
Michael De Vlieger, Diagram montage of XOR-triangles resulting from a(n) with 2 <= n <= 33.
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

Cf. A334556, A334769, A334931, A334932.

CoefficientList[Series[(1 + 11 x + 4 x^2 - 8 x^3)/(1 - 9 x^2 + 8 x^4), {x, 0, 28}], x]
(* 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 &]]

G.f.: (1 + 11*x + 4*x^2 - 8*x^3)/(1 - 9*x^2 + 8*x^4).

a(n) = - (4/7) - (1/7)*(-1)^(n-1) + ((6 + 10*sqrt(2))/7)*(2*sqrt(2))^(n-1) + ((6 - 10*sqrt(2))/7)*(-2*sqrt(2))^(n-1) - Alejandro J. Becerra Jr., May 31 2020

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.

Original entry on oeis.org

2535, 3705, 162279, 237177, 10385895, 15179385, 664697319, 971480697, 42540628455, 62174764665, 2722600221159, 3979184938617, 174246414154215, 254667836071545, 11151770505869799, 16298741508578937, 713713312375667175, 1043119456549052025, 45677651992042699239

Offset: 1

Michael De Vlieger, May 16 2020

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.

			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):
  @ . . @ @ @ @ . . @ @ @ @ . . @ @ @
   @ . @ . . . @ . @ . . . @ . @ . .
    @ @ @ . . @ @ @ @ . . @ @ @ @ .
     . . @ . @ . . . @ . @ . . . @
      . @ @ @ @ . . @ @ @ @ . . @
       @ . . . @ . @ . . . @ . @
        @ . . @ @ @ @ . . @ @ @
         @ . @ . . . @ . @ . .
          @ @ @ . . @ @ @ @ .
           . . @ . @ . . . @
            . @ @ @ @ . . @
             @ . . . @ . @
              @ . . @ @ @
               @ . @ . .
                @ @ @ .
                 . . @
                  . @
                   @

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

Cf. A334556, A334769, A334930, A334931.

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 &]]

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)

cp's OEIS Frontend

A334930 Numbers that generate rotationally symmetrical XOR-triangles featuring singleton zero bits in a hexagonal arrangement.

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