A306288 - OEIS

%I #15 Feb 06 2019 12:58:36

%S 3,0,1,2,3,3,2,3,0,0,0,1,2,1,1,2,3,3,2,3,0,3,3,2,1,2,2,3,0,3,2,3,0,0,

%T 1,0,3,0,1,0,3,0,0,1,2,1,0,1,2,2,3,2,1,1,0,1,2,1,1,2,3,3,2,3,0,3,3,2,

%U 1,2,2,3,0,3,2,3,0,0,1,0,3,3,2,3,0

%N Irregular triangle T(n,k), 1 <= n, 1 <= k <= (2/3)*(2+2^(2*n)), read by rows: T(n,k) determines absolute directions along the perimeter of the n-th T-type Hilbert Tree.

%C The T-type Hilbert trees are a sequence of polyominoes whose edges, all but one, are segments of the Hilbert curve described by A163540. One extra edge closes a loop around the perimeter (cf. Formula). The first T-type tree is a monomino with four edges, and the second is the T pentomino with 12 unit edges. All deeper trees are determined by iteration of replacement rules (cf. linked image "First Six T-type Trees"). The T-type Hilbert trees nest along the upper half plane according to the limit-periodic ruler function A001511. Such an arrangement reconstructs the Hilbert curve everywhere away from the ground axis (cf. linked image "Limit-Periodic Construction").

%H Bradley Klee, <a href="/A306288/a306288.png">Limit-Periodic Construction</a>.

%H Bradley Klee, <a href="/A306288/a306288_1.png">First Six T-type Trees</a>.

%F a(n,(2/3)*(2+2^(2*n))) = 2;

%F a(n,k) = A163540( (1/3)*(1+11*2^(2*n)+3*(-1)^n *2^(2*n+1))-1+k ).

%e T(1,k) = 3, 0, 1, 2;

%e T(2,k) = 3, 3, 2, 3, 0, 0, 0, 1, 2, 1, 1, 2.

%t HC = {L[n_ /; EvenQ[n]] :> {F[n], L[n], L[Mod[n + 1, 2]], R[n]},

%t    R[n_ /; OddQ[n]] :> {F[n], R[n], R[Mod[n + 1, 2]], L[n]},

%t    R[n_ /; EvenQ[n]] :> {L[n], R[Mod[n + 1, 2]], R[n], F[Mod[n + 1, 2]]},

%t    L[n_ /; OddQ[n]] :> {R[n], L[Mod[n + 1, 2]], L[n], F[Mod[n + 1, 2]]},

%t    F[n_ /; EvenQ[n]] :> {L[n], R[Mod[n + 1, 2]], R[n], L[Mod[n + 1, 2]]},

%t    F[n_ /; OddQ[n]] :> {R[n], L[Mod[n + 1, 2]], L[n], R[Mod[n + 1, 2]]}};

%t TurnMap = {F[_] -> 0, L[_] -> 1, R[_] -> -1};

%t T2ind[1] = 7; T2ind[2] = 27;T2ind[n_] := 5*T2ind[n - 1] - 4*T2ind[n - 2];

%t T2Vec[n_] := Append[Mod[ FoldList[Plus, Flatten[Nest[# /. HC &, F[0] /. HC, n] /.

%t        TurnMap][[T2ind[n] ;; -(T2ind[n] + 1)]]], 4], 2]

%t Flatten[T2Vec/@Range[5]]

%Y Y-Type Trees: A306287. Cf. A163540, A001511, A246559.

%K tabf,nonn

%O 1,1

%A _Bradley Klee_, Feb 03 2019