A334931 Numbers that generate rotationally symmetrical XOR-triangles with a pattern of zero-triangles of edge length 2, some of which are clipped to result in some singleton zeros at the edges.

151, 233, 1483, 1693, 10707, 13029, 644007, 941241, 317049751, 490370281, 3111314891, 3550957213, 22455577043, 27325461221, 1350581212071, 1973926386873, 664901519788951, 1028381017273577, 6524900247528907, 7446897021636253, 47092758308252115, 57305645652210405

Offset: 1

Author

Michael De Vlieger, May 16 2020

Keywords

• nonn
• easy

Comments

Subset of A334769 which is a subset of A334556.

Numbers m in this sequence A070939(m) (mod 3) = 2. The numbers in this sequence can be constructed using run lengths of bits.

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

For n = 1 (mod 8): 12..(1132)..113;

For n = 3 (mod 8): 113..(2113)..2112;

For n = 5 (mod 8): 11123..(1123)..1122;

For n = 7 (mod 8): 123112..(3112)..31123, where the parenthetic run lengths occur, when they occur, in multiples of 3. Thus, a(9) has the run length form 12113211321132113 = binary 10010111001011100101110010111 = decimal 317049751.

Examples

			Diagrams of a(1)-a(6), replacing "0" with "." and "1" with "@" for clarity:
a(1) = 151 (a(2) = 233 appears as a mirror image):
  @ . . @ . @ @ @
   @ . @ @ @ . .
    @ @ . . @ .
     . @ . @ @
      @ @ @ .
       . . @
        . @
         @
.
a(3) = 1483 (a(4) = 1693 appears as a mirror image):
  @ . @ @ @ . . @ . @ @
   @ @ . . @ . @ @ @ .
    . @ . @ @ @ . . @
     @ @ @ . . @ . @
      . . @ . @ @ @
       . @ @ @ . .
        @ . . @ .
         @ . @ @
          @ @ .
           . @
            @
.
a(5) = 10707 (a(6) = 13029 appears as a mirror image):
  @ . @ . . @ @ @ . @ . . @ @
   @ @ @ . @ . . @ @ @ . @ .
    . . @ @ @ . @ . . @ @ @
     . @ . . @ @ @ . @ . .
      @ @ . @ . . @ @ @ .
       . @ @ @ . @ . . @
        @ . . @ @ @ . @
         @ . @ . . @ @
          @ @ @ . @ .
           . . @ @ @
            . @ . .
             @ @ .
              . @
               @
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1264
• 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 sequences related to XOR-triangles
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,2097153,0,0,0,0,0,0,0,-2097152).

Crossrefs

Cf. A334556, A334769, A334930, A334932.

Programs

Mathematica

Array[FromDigits[Flatten@ MapIndexed[ConstantArray[#2, #1] & @@ {#1, Mod[First[#2], 2]} &, If[EvenQ@ #1, Reverse@ #2, #2]], 2] & @@ {#1, Which[#2 == 1, PadRight[{1, 2}, 12 Ceiling[#1/8] - 7, {3, 2, 1, 1}], #2 == 2, PadRight[{1, 1}, 12 Ceiling[#1/8] - 6, {1, 1, 3, 2}]~Join~{2}, #2 == 3, PadRight[{1, 1}, 12 Ceiling[#1/8] - 4, {3, 1, 1, 2}]~Join~{2}, True, PadRight[{}, 12 Ceiling[#1/8] - 1, {1, 2, 3, 1}]]} & @@ {#, Ceiling[Mod[#, 8]/2]} &, 22]
(* 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 G.f.: *)
Rest@ CoefficientList[Series[x (1654784 x^13 + 1359872 x^12 + 477184 x^11 + 1236992 x^10 + 1733632 x^9 + 379648 x^8 + 941241 x^7 + 644007 x^6 + 13029 x^5 + 10707 x^4 + 1693 x^3 + 1483 x^2 + 233 x + 151)/((1 - x^8) (1 - 2097152 x^8)), {x, 0, 22}], x] (* Michael De Vlieger, Mar 19 2021 *)

Formula

From Alejandro J. Becerra Jr., Mar 01 2021: (Start)

G.f.: x*(1654784*x^13 + 1359872*x^12 + 477184*x^11 + 1236992*x^10 + 1733632*x^9 + 379648*x^8 + 941241*x^7 + 644007*x^6 + 13029*x^5 + 10707*x^4 + 1693*x^3 + 1483*x^2 + 233*x + 151)/((1 - x^8)*(1 - 2097152*x^8)).

a(n) = 2097153*a(n-8) - 2097152*a(n-16). (End)