A163538 -id:A163538 - cp's OEIS frontend

A163357 Hilbert curve in N X N grid, starting rightwards from the top-left corner, listed by descending antidiagonals.

0, 1, 3, 14, 2, 4, 15, 13, 7, 5, 16, 12, 8, 6, 58, 19, 17, 11, 9, 57, 59, 20, 18, 30, 10, 54, 56, 60, 21, 23, 29, 31, 53, 55, 61, 63, 234, 22, 24, 28, 32, 52, 50, 62, 64, 235, 233, 25, 27, 35, 33, 51, 49, 67, 65, 236, 232, 230, 26, 36, 34, 46, 48, 68, 66, 78, 239, 237, 231

Offset: 0

Antti Karttunen, Jul 29 2009

			The top left 8 X 8 corner of the array shows how this surjective self-avoiding walk begins (connect the terms in numerical order, 0-1-2-3-...):
   0  1 14 15 16 19 20 21
   3  2 13 12 17 18 23 22
   4  7  8 11 30 29 24 25
   5  6  9 10 31 28 27 26
  58 57 54 53 32 35 36 37
  59 56 55 52 33 34 39 38
  60 61 50 51 46 45 40 41
  63 62 49 48 47 44 43 42

A. Karttunen, Table of n, a(n) for n = 0..32895
Jörg Arndt, Plane-filling curves on all uniform grids, arXiv:1607.02433v1 [math.CO], July 11, 2016.
Herman Haverkort, Recursive tilings and space-filling curves
Herman Haverkort, The Sound of Space-Filling Curves, Proc. Bridges 2017, pp. 399-402.
Eric Weisstein's World of Mathematics, Hilbert curve
Wikipedia, Self-avoiding walk
Wikipedia, Space-filling curve
Index entries for sequences that are permutations of the nonnegative integers

Transpose: A163359. Inverse: A163358. One-based version: A163361. Row sums: A163365. Row 0: A163482. Column 0: A163483. Central diagonal: A062880. See also A163334 & A163336 for the Peano curve.

b[{n_, k_}, {m_}] := (A[k, n] = m-1);
MapIndexed[b, List @@ HilbertCurve[4][[1]]];
Table[A[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 07 2021 *)

a(n) = A163355(A054238(n)).

Links to further derived sequences added by Antti Karttunen, Sep 21 2009

A059253 Hilbert's Hamiltonian walk on N X N projected onto y axis: m'(3).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 2, 2, 3, 4, 4, 5, 5, 6, 7, 7, 6, 6, 7, 7, 6, 5, 5, 4, 4, 4, 4, 5, 5, 6, 7, 7, 6, 6, 7, 7, 6, 5, 5, 4, 4, 3, 2, 2, 3, 3, 3, 2, 2, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 2, 3, 3, 2, 2, 3, 3, 2, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 2, 2, 3, 4, 5, 5, 4, 4, 4

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 23 2001

nonn

This is the Y-coordinate of the n-th term in the type I Hilbert's Hamiltonian walk A163359 and the X-coordinate of its transpose A163357.

Antti Karttunen, Table of n, a(n) for n = 0..65535
J. Shallit, Hilbert's spacefilling curve described by automatic, regular, and synchronized sequences, arXiv:2106.01062 [cs.FL], June 2 2021.
Index entries for sequences related to coordinates of 2D curves

See also the y-projection, m(3), A059252 as well as A163538, A163540, A163542, A059261, A059285, A163547 and A163528.

C
```
See A059252.
```

Initially [m(0) = 0, m'(0) = 0]; recursion: m(2n + 1) = m(2n).m'(2n).f(m'(2n), 2n).c(m(2n), 2n + 1); m'(2n + 1) = m'(2n).f(m(2n), 2n).f(m(2n), 2n).mir(m'(2n)); m(2n) = m(2n - 1).f(m'(2n - 1), 2n - 1).f(m'(2n - 1), 2n - 1).mir(m(2n - 1)); m'(2n) = m'(2n - 1).m(2n - 1).f(m(2n - 1), 2n - 1).c(m'(2n - 1), 2n); where f(m, n) is the alphabetic morphism i := i + 2^n [example: f(0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, 2) = 4 4 5 5 6 7 7 6 6 7 7 6 5 5 4 4]; c(m, n) is the complementation to 2^n - 1 alphabetic morphism [example: c(0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, 3) = 7 7 6 6 5 4 4 5 5 4 4 5 6 6 7 7]; and mir(m) is the mirror operator [example: mir(0 1 1 0 0 0 1 1 2 2 3 3 3 2 2 3) = 3 2 2 3 3 3 2 2 1 1 0 0 0 1 1 0].

a(n) = A025581(A163358(n)) = A002262(A163360(n)) = A059905(A163356(n)).

Extended by Antti Karttunen, Aug 01 2009

A163540 The absolute direction (0=east, 1=south, 2=west, 3=north) taken by the type I Hilbert's Hamiltonian walk A163357 at the step n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 01 2009

nonn

Taking every sixteenth term gives the same sequence: (and similarly for all higher powers of 16 as well): a(n) = a(16*n).

A. Karttunen, Table of n, a(n) for n = 1..65536

a(n) = A163540(A008598(n)) = A004442(A163541(n)). See also A163542.

HC = {L[n_ /; IntegerQ[n/2]] :> {F[n], L[n], L[n + 1], R[n + 2]},
   R[n_ /; IntegerQ[(n + 1)/2]] :> {F[n], R[n], R[n + 3], L[n + 2]},
   R[n_ /; IntegerQ[n/2]] :> {L[n], R[n + 1], R[n], F[n + 3]},
   L[n_ /; IntegerQ[(n + 1)/2]] :> {R[n], L[n + 3], L[n], F[n + 1]},
   F[n_ /; IntegerQ[n/2]] :> {L[n], R[n + 1], R[n], L[n + 3]},
   F[n_ /; IntegerQ[(n + 1)/2]] :> {R[n], L[n + 3], L[n], R[n + 1]}};
a[1] = F[0]; Map[(a[n_ /; IntegerQ[(n - #)/16]] :=
    Part[Flatten[a[(n + 16 - #)/16] /. HC /. HC], #]) &, Range[16]];
Part[FoldList[Mod[Plus[#1, #2], 4] &, 0,
  a[#] & /@ Range[4^4] /. {F[n_] :> 0, L[n_] :> 1, R[n_] :> -1}],
2 ;; -1] (* Bradley Klee, Aug 07 2015 *)

(define (A163540 n) (modulo (+ 3 (A163538 n) (A163539 n) (abs (A163539 n))) 4))

a(n) = A010873(A163538(n)+A163539(n)+abs(A163539(n))+3).

A163539 The change in Y-coordinate when moving from the n-1:th to the n-th term in the type I Hilbert's Hamiltonian walk A163357.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, -1, 0, 1, 0, -1, -1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 1, 0, 1, 0, 1, 0, 0, -1, 0, 1, 1, 1, 0, -1, 0, 0, 1, 0, 1, 0, 1, 0, 0, -1, 0, 1, 0, 0, -1, 0, -1, -1, 0, 1, 0, -1, 0, 1, 1, 0, 1, 0, 1, 1, 0, -1, 0, 0, 1, 0, 1, 0, 1, 0, 0, -1, 0, 1, 1, 0, 1, 0, 1, 1, 0, -1, 0, 1, 0, -1, -1, 0

Offset: 0

Antti Karttunen, Aug 01 2009

sign

A. Karttunen, Table of n, a(n) for n = 0..65536

These are the first differences of A059252. See also: A163538, A163541, A163543.

a(0)=0, a(n) = A059252(n) - A059252(n-1).

A163541 The absolute direction (0=east, 1=south, 2=west, 3=north) taken by the type I Hilbert's Hamiltonian walk A163359 at the step n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 01 2009

nonn

Taking every sixteenth term gives the same sequence: (and similarly for all higher powers of 16 as well): a(n) = a(16*n).

Antti Karttunen, Table of n, a(n) for n = 1..4096

a(n) = A163541(A008598(n)) = A004442(A163540(n)). See also A163543.

HC = {L[n_ /; IntegerQ[n/2]] :> {F[n], L[n], L[n + 1], R[n + 2]},
R[n_ /; IntegerQ[(n + 1)/2]] :> {F[n], R[n], R[n + 3], L[n + 2]},
R[n_ /; IntegerQ[n/2]] :> {L[n], R[n + 1], R[n], F[n + 3]},
L[n_ /; IntegerQ[(n + 1)/2]] :> {R[n], L[n + 3], L[n], F[n + 1]},
F[n_ /; IntegerQ[n/2]] :> {L[n], R[n + 1], R[n], L[n + 3]},
F[n_ /; IntegerQ[(n + 1)/2]] :> {R[n], L[n + 3], L[n], R[n + 1]}};
a[1] = L[0]; Map[(a[n_ /; IntegerQ[(n - #)/16]] := Part[Flatten[a[(n + 16 - #)/16]/.HC/.HC],#]) &, Range[16]];
Part[FoldList[Mod[Plus[#1, #2], 4] &, 0, a[#] & /@ Range[4^4]/.{F[n_]:>0,L[n_]:>1,R[n_]:>-1}], 2 ;; -1] (* Bradley Klee, Aug 07 2015 *)

(define (A163541 n) (modulo (+ 3 (A163538 n) (A163539 n) (abs (A163538 n))) 4))

a(n) = A010873(A163538(n) + A163539(n) + abs(A163538(n)) + 3).

cp's OEIS Frontend

A163357 Hilbert curve in N X N grid, starting rightwards from the top-left corner, listed by descending antidiagonals.

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A059253 Hilbert's Hamiltonian walk on N X N projected onto y axis: m'(3).

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A163540 The absolute direction (0=east, 1=south, 2=west, 3=north) taken by the type I Hilbert's Hamiltonian walk A163357 at the step n.

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A163539 The change in Y-coordinate when moving from the n-1:th to the n-th term in the type I Hilbert's Hamiltonian walk A163357.

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A163541 The absolute direction (0=east, 1=south, 2=west, 3=north) taken by the type I Hilbert's Hamiltonian walk A163359 at the step n.

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