A334771 a(n) = smallest m that generates a rotationally symmetrical XOR-triangle T(m) with a central triangle of zeros with side length n.

543, 151, 2359, 599, 8607, 2391, 37687, 9559, 137631, 38231, 602935, 152919, 2202015, 611671, 9646903, 2446679, 35232159, 9786711, 154350391, 39146839, 563714463, 156587351, 2469606199, 626349399, 9019431327, 2505397591, 39513699127, 10021590359

Offset: 1

Author

Michael De Vlieger, May 10 2020

Keywords

• nonn
• easy

Comments

An XOR-triangle T(m) is an inverted 0-1 triangle formed by choosing a top row the binary rendition of n and having each entry in subsequent rows be the XOR of the two values above it, i.e., A038554(m) applied recursively until we reach a single bit.

A334556 is the sequence of rotationally symmetrical T(m).

A central zero-triangle (CZT) is a field of contiguous 0-bits with side length n in T(m) surrounded on all sides by a layer of 1 bits, and generally k > 1 bits of any parity. Alternatively, these might be referred to as "central bubbles".

Examples

			First 4 terms shown below, replacing 0 with “.” for clarity:
    a(1) = 543; T(543):
  1 . . . . 1 1 1 1 1
   1 . . . 1 . . . .
    1 . . 1 1 . . .
     1 . 1 . 1 . .
      1 1 1 1 1 .
       . . . . 1
        . . . 1
         . . 1
          . 1
           1
a(2) = 151; T(151):
  1 . . 1 . 1 1 1
   1 . 1 1 1 . .
    1 1 . . 1 .
     . 1 . 1 1
      1 1 1 .
       . . 1
        . 1
         1
a(3) = 2359; T(2359):
  1 . . 1 . . 1 1 . 1 1 1
   1 . 1 1 . 1 . 1 1 . .
    1 1 . 1 1 1 1 . 1 .
     . 1 1 . . . 1 1 1
      1 . 1 . . 1 . .
       1 1 1 . 1 1 .
        . . 1 1 . 1
         . 1 . 1 1
          1 1 1 .
           . . 1
            . 1
             1
a(4) = 599; T(599):
  1 . . 1 . 1 . 1 1 1
   1 . 1 1 1 1 1 . .
    1 1 . . . . 1 .
     . 1 . . . 1 1
      1 1 . . 1 .
       . 1 . 1 1
        1 1 1 .
         . . 1
          . 1
           1
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..3314
• Michael De Vlieger, Central zero-triangles in rotationally symmetrical XOR-Triangles, 2020.
• Michael De Vlieger, Diagram montage of XOR-triangles of the first 64 terms.
• Michael De Vlieger, Correlation of A334771, A334769, A334556, and A333624.
• Index entries for sequences related to binary expansion of n
• Index entries for sequences related to XOR-triangles
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,17,0,0,0,-16).

Crossrefs

Cf. A038554, A070939, A334556, A334769, A334770.

Programs

Mathematica

Block[{f, s = Rest[Import["https://oeis.org/A334556/b334556.txt", "Data"][[All, -1]] ], t, u}, f[n_] := NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[n, 2], Length@ # > 1 &]; Set[{t, u}, Transpose@ Array[Block[{n = s[[#]]}, If[# == 0, Nothing, {n, #}] &@ FirstCase[MapIndexed[If[2 #2 > #3 + 1, Nothing, #1[[#2 ;; -#2]]] & @@ {#1, First[#2], Length@ #1} &, f[n][[1 ;; Ceiling[IntegerLength[#, 2]/(2 Sqrt[3])] + 3]]  ], r_List /; FreeQ[r, 1] :> Length@ r] /. k_ /; MissingQ@ k -> 0] &, Length@ s - 1, 2]]; Array[If[! IntegerQ@ #, 0, t[[#]] ] &@ FirstPosition[u, #][[1]] &, Max@ u] ]
(* Second, more efficient program: *)
LinearRecurrence[{0, 0, 0, 17, 0, 0, 0, -16}, {543, 151, 2359, 599, 8607, 2391, 37687, 9559}, 28] (* Michael De Vlieger, May 20 2020 *)

PARI

Vec(x*(543 + 151*x + 2359*x^2 + 599*x^3 - 624*x^4 - 176*x^5 - 2416*x^6 - 624*x^7) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 + x^2)*(1 + 4*x^2)) + O(x^30)) \\ Colin Barker, May 21 2020

Formula

a(n) = 17*a(n-4) - 16*a(n-8), starting with a(1) = 543, a(2) = 151, a(3) = 2359, a(4) = 599, a(5) = 8607, a(6) = 2391, a(7) = 37687, and a(8) = 9559.

G.f.: x*(543 + 151*x + 2359*x^2 + 599*x^3 - 624*x^4 - 176*x^5 - 2416*x^6 - 624*x^7) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 + x^2)*(1 + 4*x^2)). - Colin Barker, May 21 2020