A306288 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.

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, 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, 1, 2, 2, 3, 0, 3, 2, 3, 0, 0, 1, 0, 3, 3, 2, 3, 0

Offset

1,1

Comments

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").

Links

Table of n, a(n) for n=1..85.

Bradley Klee, Limit-Periodic Construction.

Bradley Klee, First Six T-type Trees.

Formula

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

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

Example

T(1,k) = 3, 0, 1, 2;
T(2,k) = 3, 3, 2, 3, 0, 0, 0, 1, 2, 1, 1, 2.

Mathematica

HC = {L[n_ /; EvenQ[n]] :> {F[n], L[n], L[Mod[n + 1, 2]], R[n]},
   R[n_ /; OddQ[n]] :> {F[n], R[n], R[Mod[n + 1, 2]], L[n]},
   R[n_ /; EvenQ[n]] :> {L[n], R[Mod[n + 1, 2]], R[n], F[Mod[n + 1, 2]]},
   L[n_ /; OddQ[n]] :> {R[n], L[Mod[n + 1, 2]], L[n], F[Mod[n + 1, 2]]},
   F[n_ /; EvenQ[n]] :> {L[n], R[Mod[n + 1, 2]], R[n], L[Mod[n + 1, 2]]},
   F[n_ /; OddQ[n]] :> {R[n], L[Mod[n + 1, 2]], L[n], R[Mod[n + 1, 2]]}};
TurnMap = {F[_] -> 0, L[_] -> 1, R[_] -> -1};
T2ind[1] = 7; T2ind[2] = 27; T2ind[n_] := 5*T2ind[n - 1] - 4*T2ind[n - 2];
T2Vec[n_] := Append[Mod[ FoldList[Plus, Flatten[Nest[# /. HC &, F[0] /. HC, n] /.
       TurnMap][[T2ind[n] ;; -(T2ind[n] + 1)]]], 4], 2]
Flatten[T2Vec/@Range[5]]

Crossrefs

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

Sequence in context: A136748 A235919 A229654 * A272188 A049765 A343394

Adjacent sequences: A306285 A306286 A306287 * A306289 A306290 A306291

Keyword

tabf,nonn

Author

Bradley Klee, Feb 03 2019

Status

approved