A163359 -id:A163359 - cp's OEIS frontend

A163360 Inverse permutation to A163359.

0, 2, 4, 1, 3, 6, 11, 7, 12, 17, 24, 18, 13, 8, 5, 9, 14, 19, 26, 20, 27, 35, 43, 34, 42, 52, 62, 51, 41, 33, 25, 32, 40, 49, 60, 50, 61, 73, 85, 72, 84, 98, 112, 97, 83, 71, 59, 70, 58, 47, 38, 48, 39, 31, 23, 30, 22, 16, 10, 15, 21, 29, 37, 28, 36, 45, 56, 46, 57, 69, 81

Offset: 0

Antti Karttunen, Jul 29 2009

abs(A025581(a(n+1)) - A025581(a(n))) + abs(A002262(a(n+1)) - A002262(a(n))) = 1 for all n.

Inverse: A163359. a(n) = A061579(A163358(n)). One-based version: A163364.

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).

A163543 The relative direction (0=straight ahead, 1=turn right, 2=turn left) taken by the type I Hilbert's Hamiltonian walk A163359 at the step n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 01 2009

nonn

a(16*n) = a(256*n) for all n.

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

a(n) = A014681(A163542(n)). See also A163541.

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[a[#] & /@ Range[4^4] /. {L[] -> 2, R[] -> 1, F[] -> 0}, 2 ;; -1] (* _Bradley Klee, Aug 06 2015 *)

(define (A163543 n) (A163241 (modulo (- (A163541 (1+ n)) (A163541 n)) 4)))

a(n) = A163241((A163541(n+1)-A163541(n)) modulo 4).

A165466 Squared distance between n's location in A163334 array and A163359 array.

Original entry on oeis.org

0, 2, 2, 2, 2, 10, 10, 2, 0, 0, 2, 10, 20, 10, 10, 18, 32, 32, 50, 74, 100, 100, 72, 50, 32, 50, 50, 34, 20, 20, 16, 16, 16, 10, 4, 4, 2, 4, 8, 8, 8, 10, 20, 18, 20, 20, 26, 50, 50, 40, 20, 20, 20, 20, 32, 32, 34, 40, 58, 74, 100, 74, 74, 80, 80, 80, 52, 52, 50, 34, 34, 32

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

Equivalently, squared distance between n's location in A163336 array and A163357 array. See example at A166043.

A. Karttunen, Table of n, a(n) for n = 0..65535

Positions of zeros: A165467. See also A166043, A165464, A163897, A163900.

a(n) = A000290(abs(A163529(n)-A059253(n))) + A000290(abs(A163528(n)-A059252(n))).

A163365 Row sums of A163357 and A163359.

Original entry on oeis.org

0, 4, 20, 40, 100, 172, 248, 336, 568, 820, 1100, 1400, 1692, 2012, 2352, 2720, 3632, 4580, 5572, 6600, 7700, 8844, 10024, 11248, 12392, 13588, 14844, 16152, 17484, 18876, 20320, 21824, 25440, 29124, 32884, 36712, 40644, 44652, 48728, 52880

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

All terms seem to be divisible by 4. Cf. A163477.

Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 42-43.

Cf. A163357, A163359.

nn = 7; s[{n_, k_}, {m_}] := (a[k, n] = m - 1); MapIndexed[s, List @@ HilbertCurve[nn][[1]]]; Total /@ Table[a[n - k, k], {n, 0, nn^2}, {k, n, 0, -1}] (* Michael De Vlieger, Nov 01 2022, after Jean-François Alcover at A163357 *)

A163482 Row 0 of A163357 (column 0 of A163359).

Original entry on oeis.org

0, 1, 14, 15, 16, 19, 20, 21, 234, 235, 236, 239, 240, 241, 254, 255, 256, 259, 260, 261, 314, 315, 316, 319, 320, 321, 334, 335, 336, 339, 340, 341, 3754, 3755, 3756, 3759, 3760, 3761, 3774, 3775, 3776, 3779, 3780, 3781, 3834, 3835, 3836, 3839

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

Cf. A163483.

A163483 Row 0 of A163359 (column 0 of A163357).

Original entry on oeis.org

0, 3, 4, 5, 58, 59, 60, 63, 64, 65, 78, 79, 80, 83, 84, 85, 938, 939, 940, 943, 944, 945, 958, 959, 960, 963, 964, 965, 1018, 1019, 1020, 1023, 1024, 1025, 1038, 1039, 1040, 1043, 1044, 1045, 1258, 1259, 1260, 1263, 1264, 1265, 1278, 1279, 1280, 1283

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

Cf. A163482.

A163477 Row sums of A163357 and A163359 divided by 4.

Original entry on oeis.org

0, 1, 5, 10, 25, 43, 62, 84, 142, 205, 275, 350, 423, 503, 588, 680, 908, 1145, 1393, 1650, 1925, 2211, 2506, 2812, 3098, 3397, 3711, 4038, 4371, 4719, 5080, 5456, 6360, 7281, 8221, 9178, 10161, 11163, 12182, 13220, 14310, 15421, 16555, 17710

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 42-43.

Cf. A163357, A163359.

nn = 8; s[{n_, k_}, {m_}] := (a[k, n] = m - 1); MapIndexed[s, List @@ HilbertCurve[nn][[1]]]; Floor[1/4*Map[Total, Table[a[n - k, k], {n, 0, nn^2}, {k, n, 0, -1}]]] (* Michael De Vlieger, Nov 01 2022, after Jean-François Alcover at A163357 *)

a(n) = floor(A163365(n)/4) (floor probably unnecessary).

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

Original entry on oeis.org

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

A163334 Peano curve in an n X n grid, starting rightwards from the top left corner, listed antidiagonally as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ... .

Original entry on oeis.org

0, 1, 5, 2, 4, 6, 15, 3, 7, 47, 16, 14, 8, 46, 48, 17, 13, 9, 45, 49, 53, 18, 12, 10, 44, 50, 52, 54, 19, 23, 11, 43, 39, 51, 55, 59, 20, 22, 24, 42, 40, 38, 56, 58, 60, 141, 21, 25, 29, 41, 37, 69, 57, 61, 425, 142, 140, 26, 28, 30, 36, 70, 68, 62, 424, 426, 143, 139

Offset: 0

Antti Karttunen, Jul 29 2009

			The top left 9 X 9 corner of the array shows how this surjective self-avoiding walk begins (connect the terms in numerical order, 0-1-2-3-...):
   0  1  2 15 16 17 18 19 20
   5  4  3 14 13 12 23 22 21
   6  7  8  9 10 11 24 25 26
  47 46 45 44 43 42 29 28 27
  48 49 50 39 40 41 30 31 32
  53 52 51 38 37 36 35 34 33
  54 55 56 69 70 71 72 73 74
  59 58 57 68 67 66 77 76 75
  60 61 62 63 64 65 78 79 80

Antti Karttunen, Table of n, a(n) for n = 0..13202
E. H. Moore, On Certain Crinkly Curves, Transactions of the American Mathematical Society, volume 1, number 1, 1900, pages 72-90. (And errata.) See section 7 (and in figure 3 rotate -90 degrees for the table here).
Giuseppe Peano, Sur une courbe, qui remplit toute une aire plane, Mathematische Annalen, volume 36, number 1, 1890, pages 157-160. Also EUDML (link to GDZ).
Eric Weisstein's World of Mathematics, Hilbert curve (this curve called "Hilbert II").
Wikipedia, Self-avoiding walk
Wikipedia, Space-filling curve
Index entries for sequences that are permutations of the natural numbers

Transpose: A163336. Inverse: A163335. One-based version: A163338. Row sums: A163342. Row 0: A163480. Column 0: A163481. Central diagonal: A163343.
See also: A163528-A163529, A163531, A163534, A163536, A163897.
See A163357 and A163359 for the Hilbert curve.

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

a(n) = A163332(A163328(n)).

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

Name corrected by Kevin Ryde, Aug 22 2020

cp's OEIS Frontend

A163360 Inverse permutation to A163359.

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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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A163543 The relative direction (0=straight ahead, 1=turn right, 2=turn left) taken by the type I Hilbert's Hamiltonian walk A163359 at the step n.

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A165466 Squared distance between n's location in A163334 array and A163359 array.

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A163365 Row sums of A163357 and A163359.

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A163482 Row 0 of A163357 (column 0 of A163359).

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A163483 Row 0 of A163359 (column 0 of A163357).

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A163477 Row sums of A163357 and A163359 divided by 4.

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

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A163334 Peano curve in an n X n grid, starting rightwards from the top left corner, listed antidiagonally as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ... .

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