

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 nth Ytype 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
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OFFSET

1,3


COMMENTS

The Ytype 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 Ytype 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 Ytype Trees"). The Ytype Hilbert trees nest along the upper half plane according to the limitperiodic ruler function A001511. Such an arrangement reconstructs the Hilbert curve everywhere away from the ground axis (cf. linked image "LimitPeriodic Construction").


LINKS



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



KEYWORD

tabf,nonn


AUTHOR



STATUS

approved



