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

Author

Michael De Vlieger, May 16 2020

Keywords

• nonn
• easy

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.

Examples

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

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

Crossrefs

Cf. A334556, A334769, A334931, A334932.

Programs

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 &]]

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