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

1,2

Comments

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.

Links

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

Formula

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

Example

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
  @ . @ @ . @ @      @ @ . @ @ . @
   @ @ . @ @ .        . @ @ . @ @
    . @ @ . @          @ . @ @ .
     @ . @ @            @ @ . @
      @ @ .              . @ @
       . @                @ .
        @                  @

Mathematica

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 &]]

Crossrefs

Cf. A334556, A334769, A334931, A334932.

Sequence in context: A056446 A023317 A056436 * A089824 A276694 A086549

Adjacent sequences: A334927 A334928 A334929 * A334931 A334932 A334933

Keyword

nonn,easy

Author

Michael De Vlieger, May 16 2020

Status

approved