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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
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: A245188 A137241 A331539 * A016457 A181715 A077089
KEYWORD
tabf,nonn
AUTHOR
Bradley Klee, Feb 03 2019
STATUS
approved

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Last modified June 13 17:32 EDT 2024. Contains 373391 sequences. (Running on oeis4.)