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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.

%I A334932 #17 Jun 21 2020 06:03:42
%S A334932 2535,3705,162279,237177,10385895,15179385,664697319,971480697,
%T A334932 42540628455,62174764665,2722600221159,3979184938617,174246414154215,
%U A334932 254667836071545,11151770505869799,16298741508578937,713713312375667175,1043119456549052025,45677651992042699239
%N 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.
%C A334932 Subset of A334769 which is a subset of A334556.
%C A334932 Numbers m in this sequence A070939(m) (mod 3) = 0. All m have first and last bits = 1.
%C A334932 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.
%C A334932 Thus, the smallest number m has run lengths {1, 2, 4, 2, 3}, which is the binary 100111100111 = decimal 2535.
%C A334932 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.
%H A334932 Michael De Vlieger, <a href="/A334932/b334932.txt">Table of n, a(n) for n = 1..1104</a>
%H A334932 Michael De Vlieger, <a href="/A334932/a334932.png">Diagram montage</a> of XOR-triangles resulting from a(n) with 1 <= n <= 32.
%H A334932 Michael De Vlieger, <a href="http://vincico.com/seq/a334769.html">Central zero-triangles in rotationally symmetrical XOR-Triangles</a>, 2020.
%H A334932 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A334932 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,65,0,-64).
%H A334932 <a href="/index/X#XOR-triangles">Index entries for sequences related to XOR-triangles</a>
%F A334932 From _Colin Barker_, Jun 09 2020: (Start)
%F A334932 G.f.: 3*x*(13 + 19*x)*(65 - 64*x^2) / ((1 - x)*(1 + x)*(1 - 8*x)*(1 + 8*x)).
%F A334932 a(n) = 65*a(n-2) - 64*a(n-4) for n>4.
%F A334932 a(n) = (1/21)*(-16 - 3*(-1)^n + 123*2^(5+3*n) - 85*(-1)^n*2^(5 + 3*n)) for n>0.
%F A334932 (End)
%e A334932 Diagrams of a(1)-a(4), replacing “0” with “.” and “1” with “@” for clarity:
%e A334932 a(1) = 2535 (a(2) = 3705 appears as a mirror image):
%e A334932   @ . . @ @ @ @ . . @ @ @
%e A334932    @ . @ . . . @ . @ . .
%e A334932     @ @ @ . . @ @ @ @ .
%e A334932      . . @ . @ . . . @
%e A334932       . @ @ @ @ . . @
%e A334932        @ . . . @ . @
%e A334932         @ . . @ @ @
%e A334932          @ . @ . .
%e A334932           @ @ @ .
%e A334932            . . @
%e A334932             . @
%e A334932              @
%e A334932 .
%e A334932 a(3) = 162279 (a(4) = 237177 appears as a mirror image):
%e A334932   @ . . @ @ @ @ . . @ @ @ @ . . @ @ @
%e A334932    @ . @ . . . @ . @ . . . @ . @ . .
%e A334932     @ @ @ . . @ @ @ @ . . @ @ @ @ .
%e A334932      . . @ . @ . . . @ . @ . . . @
%e A334932       . @ @ @ @ . . @ @ @ @ . . @
%e A334932        @ . . . @ . @ . . . @ . @
%e A334932         @ . . @ @ @ @ . . @ @ @
%e A334932          @ . @ . . . @ . @ . .
%e A334932           @ @ @ . . @ @ @ @ .
%e A334932            . . @ . @ . . . @
%e A334932             . @ @ @ @ . . @
%e A334932              @ . . . @ . @
%e A334932               @ . . @ @ @
%e A334932                @ . @ . .
%e A334932                 @ @ @ .
%e A334932                  . . @
%e A334932                   . @
%e A334932                    @
%t A334932 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]
%t A334932 (* Generate a textual plot of XOR-triangle T(n) *)
%t A334932 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 &]]
%Y A334932 Cf. A334556, A334769, A334930, A334931.
%K A334932 nonn,easy
%O A334932 1,1
%A A334932 _Michael De Vlieger_, May 16 2020