A334930 -id:A334930 - cp's OEIS frontend

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

Michael De Vlieger, May 16 2020

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.

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

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).

Cf. A334556, A334769, A334930, A334932.

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 *)

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)

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

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.

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