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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.

%I A334931 #19 Mar 24 2021 23:08:47
%S A334931 151,233,1483,1693,10707,13029,644007,941241,317049751,490370281,
%T A334931 3111314891,3550957213,22455577043,27325461221,1350581212071,
%U A334931 1973926386873,664901519788951,1028381017273577,6524900247528907,7446897021636253,47092758308252115,57305645652210405
%N 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.
%C A334931 Subset of A334769 which is a subset of A334556.
%C A334931 Numbers m in this sequence A070939(m) (mod 3) = 2. The numbers in this sequence can be constructed using run lengths of bits.
%C A334931 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.
%C A334931 For n = 1 (mod 8): 12..(1132)..113;
%C A334931 For n = 3 (mod 8): 113..(2113)..2112;
%C A334931 For n = 5 (mod 8): 11123..(1123)..1122;
%C A334931 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.
%H A334931 Michael De Vlieger, <a href="/A334931/b334931.txt">Table of n, a(n) for n = 1..1264</a>
%H A334931 Michael De Vlieger, <a href="/A334931/a334931.png">Diagram montage</a> of XOR-triangles resulting from a(n) with 1 <= n <= 32.
%H A334931 Michael De Vlieger, <a href="http://vincico.com/seq/a334769.html">Central zero-triangles in rotationally symmetrical XOR-Triangles</a>, 2020.
%H A334931 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A334931 <a href="/index/X#XOR-triangles">Index entries for sequences related to XOR-triangles</a>
%H A334931 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,2097153,0,0,0,0,0,0,0,-2097152).
%F A334931 From _Alejandro J. Becerra Jr._, Mar 01 2021: (Start)
%F A334931 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)).
%F A334931 a(n) = 2097153*a(n-8) - 2097152*a(n-16). (End)
%e A334931 Diagrams of a(1)-a(6), replacing "0" with "." and "1" with "@" for clarity:
%e A334931 a(1) = 151 (a(2) = 233 appears as a mirror image):
%e A334931   @ . . @ . @ @ @
%e A334931    @ . @ @ @ . .
%e A334931     @ @ . . @ .
%e A334931      . @ . @ @
%e A334931       @ @ @ .
%e A334931        . . @
%e A334931         . @
%e A334931          @
%e A334931 .
%e A334931 a(3) = 1483 (a(4) = 1693 appears as a mirror image):
%e A334931   @ . @ @ @ . . @ . @ @
%e A334931    @ @ . . @ . @ @ @ .
%e A334931     . @ . @ @ @ . . @
%e A334931      @ @ @ . . @ . @
%e A334931       . . @ . @ @ @
%e A334931        . @ @ @ . .
%e A334931         @ . . @ .
%e A334931          @ . @ @
%e A334931           @ @ .
%e A334931            . @
%e A334931             @
%e A334931 .
%e A334931 a(5) = 10707 (a(6) = 13029 appears as a mirror image):
%e A334931   @ . @ . . @ @ @ . @ . . @ @
%e A334931    @ @ @ . @ . . @ @ @ . @ .
%e A334931     . . @ @ @ . @ . . @ @ @
%e A334931      . @ . . @ @ @ . @ . .
%e A334931       @ @ . @ . . @ @ @ .
%e A334931        . @ @ @ . @ . . @
%e A334931         @ . . @ @ @ . @
%e A334931          @ . @ . . @ @
%e A334931           @ @ @ . @ .
%e A334931            . . @ @ @
%e A334931             . @ . .
%e A334931              @ @ .
%e A334931               . @
%e A334931                @
%t A334931 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]
%t A334931 (* Generate a textual plot of XOR-triangle T(n) *)
%t A334931 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 &]]
%t A334931 (* From G.f.: *)
%t A334931 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 *)
%Y A334931 Cf. A334556, A334769, A334930, A334932.
%K A334931 nonn,easy
%O A334931 1,1
%A A334931 _Michael De Vlieger_, May 16 2020