A306287 - OEIS

%I #16 Feb 06 2019 12:58:15

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

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

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

%N Irregular triangle T(n,k), 1 <= n, 1 <= k <= (1/6)*(4+5*2^(2*n)), read by rows: T(n,k) determines absolute directions along the perimeter of the n-th Y-type Hilbert Tree.

%C The Y-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 Y-type tree is a monomino with four edges, and the second is the Y hexomino with 14 unit edges. All deeper trees are determined by iteration of replacement rules (cf. linked image "First Six Y-type Trees"). The Y-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="/A306287/a306287.png">Limit-Periodic Construction</a>.

%H Bradley Klee, <a href="/A306287/a306287_1.png">First Six Y-type Trees</a>.

%F a(n,(1/6)*(4+5*2^(2*n))) = 2;

%F a(n,k) = A163540( (1/12)*(8+7*2^(2*n)-3*(-1)^n *2^(2*n+1))-1+k ).

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

%e T(2,k) = 1, 2, 1, 1, 0, 3, 0, 1, 0, 3, 3, 2, 3, 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 T1ind[1] = 1; T1ind[2] = 2; T1ind[n_] := 5*T1ind[n - 1] - 4*T1ind[n - 2];

%t T1Vec[n_] := Append[Mod[FoldList[Plus, Flatten[Nest[# /. HC &, F[0],

%t         n] /. TurnMap][[T1ind[n] ;; -(T1ind[n] + 1)]]], 4], 2]

%t Flatten[T1Vec /@ Range[5]]

%Y T-Type Trees: A306288. Cf. A163540, A001511, A246559.

%K tabf,nonn

%O 1,3

%A _Bradley Klee_, Feb 03 2019