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

%I A334771 #24 May 21 2020 16:52:07
%S A334771 543,151,2359,599,8607,2391,37687,9559,137631,38231,602935,152919,
%T A334771 2202015,611671,9646903,2446679,35232159,9786711,154350391,39146839,
%U A334771 563714463,156587351,2469606199,626349399,9019431327,2505397591,39513699127,10021590359
%N A334771 a(n) = smallest m that generates a rotationally symmetrical XOR-triangle T(m) with a central triangle of zeros with side length n.
%C A334771 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.
%C A334771 A334556 is the sequence of rotationally symmetrical T(m).
%C A334771 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".
%H A334771 Michael De Vlieger, <a href="/A334771/b334771.txt">Table of n, a(n) for n = 1..3314</a>
%H A334771 Michael De Vlieger, <a href="http://vincico.com/seq/a334769.html">Central zero-triangles in rotationally symmetrical XOR-Triangles</a>, 2020.
%H A334771 Michael De Vlieger, <a href="/A334771/a334771_1.png">Diagram montage</a> of XOR-triangles of the first 64 terms.
%H A334771 Michael De Vlieger, <a href="/A334771/a334771.txt">Correlation of A334771, A334769, A334556, and A333624</a>.
%H A334771 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A334771 <a href="/index/X#XOR-triangles">Index entries for sequences related to XOR-triangles</a>
%H A334771 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,17,0,0,0,-16).
%F A334771 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.
%F A334771 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
%e A334771 First 4 terms shown below, replacing 0 with “.” for clarity:
%e A334771     a(1) = 543; T(543):
%e A334771   1 . . . . 1 1 1 1 1
%e A334771    1 . . . 1 . . . .
%e A334771     1 . . 1 1 . . .
%e A334771      1 . 1 . 1 . .
%e A334771       1 1 1 1 1 .
%e A334771        . . . . 1
%e A334771         . . . 1
%e A334771          . . 1
%e A334771           . 1
%e A334771            1
%e A334771 a(2) = 151; T(151):
%e A334771   1 . . 1 . 1 1 1
%e A334771    1 . 1 1 1 . .
%e A334771     1 1 . . 1 .
%e A334771      . 1 . 1 1
%e A334771       1 1 1 .
%e A334771        . . 1
%e A334771         . 1
%e A334771          1
%e A334771 a(3) = 2359; T(2359):
%e A334771   1 . . 1 . . 1 1 . 1 1 1
%e A334771    1 . 1 1 . 1 . 1 1 . .
%e A334771     1 1 . 1 1 1 1 . 1 .
%e A334771      . 1 1 . . . 1 1 1
%e A334771       1 . 1 . . 1 . .
%e A334771        1 1 1 . 1 1 .
%e A334771         . . 1 1 . 1
%e A334771          . 1 . 1 1
%e A334771           1 1 1 .
%e A334771            . . 1
%e A334771             . 1
%e A334771              1
%e A334771 a(4) = 599; T(599):
%e A334771   1 . . 1 . 1 . 1 1 1
%e A334771    1 . 1 1 1 1 1 . .
%e A334771     1 1 . . . . 1 .
%e A334771      . 1 . . . 1 1
%e A334771       1 1 . . 1 .
%e A334771        . 1 . 1 1
%e A334771         1 1 1 .
%e A334771          . . 1
%e A334771           . 1
%e A334771            1
%t A334771 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] ]
%t A334771 (* Second, more efficient program: *)
%t A334771 LinearRecurrence[{0, 0, 0, 17, 0, 0, 0, -16}, {543, 151, 2359, 599, 8607, 2391, 37687, 9559}, 28] (* _Michael De Vlieger_, May 20 2020 *)
%o A334771 (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
%Y A334771 Cf. A038554, A070939, A334556, A334769, A334770.
%K A334771 nonn,easy
%O A334771 1,1
%A A334771 _Michael De Vlieger_, May 10 2020