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A306287 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. 3
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, 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, 3, 2, 3, 2, 1, 2, 1, 0, 1, 2, 2, 3, 2, 1, 1, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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").

LINKS

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

Bradley Klee, Limit-Periodic Construction.

Bradley Klee, First Six Y-type Trees.

FORMULA

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

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

EXAMPLE

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

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

T1ind[1] = 1; T1ind[2] = 2; T1ind[n_] := 5*T1ind[n - 1] - 4*T1ind[n - 2];

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

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

Flatten[T1Vec /@ Range[5]]

CROSSREFS

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

Sequence in context: A136170 A245188 A137241 * A016457 A181715 A077089

Adjacent sequences:  A306284 A306285 A306286 * A306288 A306289 A306290

KEYWORD

tabf,nonn

AUTHOR

Bradley Klee, Feb 03 2019

STATUS

approved

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Last modified January 27 18:36 EST 2020. Contains 331296 sequences. (Running on oeis4.)