A163334 -id:A163334 - cp's OEIS frontend

A163481 Row 0 of A163336 (column 0 of A163334).

0, 5, 6, 47, 48, 53, 54, 59, 60, 425, 426, 431, 432, 437, 438, 479, 480, 485, 486, 491, 492, 533, 534, 539, 540, 545, 546, 3827, 3828, 3833, 3834, 3839, 3840, 3881, 3882, 3887, 3888, 3893, 3894, 3935, 3936, 3941, 3942, 3947, 3948, 4313, 4314, 4319

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

Cf. A163480 (X axis), A208665 (Z-order Y axis).

a(n) = my(v=digits(n,3),s=Mod(0,2)); for(i=1,#v, s+=v[i]; v[i]=3*v[i]+if(s,2)); fromdigits(v,9); \\ Kevin Ryde, Oct 06 2020

From Kevin Ryde, Oct 06 2020: (Start)
a(n) = A163332(A208665(n)), including at n=0 by reckoning A208665(0)=0.
a(n) = 3*A163480(n) + (2 if n odd).
(End)

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))).

A166042 Permutation of nonnegative integers: a(n) tells which integer is in the same position in the square array A163334 as where n is located in the array A163357.

Original entry on oeis.org

0, 1, 4, 5, 6, 47, 46, 7, 8, 45, 44, 9, 14, 3, 2, 15, 16, 13, 12, 17, 18, 19, 22, 23, 24, 25, 28, 29, 42, 11, 10, 43, 40, 37, 36, 41, 30, 31, 34, 35, 72, 73, 76, 77, 66, 71, 70, 67, 68, 57, 56, 69, 38, 39, 50, 51, 52, 49, 48, 53, 54, 55, 58, 59, 60, 425, 424, 61, 62, 63, 422

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

			The top left 8 X 8 corner of A163357:
   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
The top left 9 X 9 corner of A163334:
   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
9 is in position (3,2) in A163357, while A163334(3,2) = 45. Thus a(9) = 45.

Inverse: A166041. a(n) = A163334(A163358(n)) = A163336(A163360(n)). Fixed points: A165465. Cf. also A166044.

A163342 Row sums of A163334 and A163336.

Original entry on oeis.org

0, 6, 12, 72, 132, 186, 240, 300, 360, 906, 1452, 1992, 2532, 3078, 3624, 4116, 4608, 5094, 5580, 6072, 6564, 7110, 7656, 8196, 8736, 9282, 9828, 14748, 19668, 24582, 29496, 34416, 39336, 44202, 49068, 53928, 58788, 63654, 68520, 73440, 78360

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

All terms seem to be divisible by 6. Cf. A163479.

A163343 Central diagonal of A163334 and A163336.

Original entry on oeis.org

0, 4, 8, 44, 40, 36, 72, 76, 80, 404, 400, 396, 360, 364, 368, 332, 328, 324, 648, 652, 656, 692, 688, 684, 720, 724, 728, 3644, 3640, 3636, 3600, 3604, 3608, 3572, 3568, 3564, 3240, 3244, 3248, 3284, 3280, 3276, 3312, 3316, 3320, 2996, 2992, 2988

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

It is easy to see by induction that these terms are always divisible by 4.

Cf. A163344 (quarter), A128173, A163332, A163334, A338086.

Peano curve axes: A163480, A163481.

a(n) = my(v=digits(n,3),s=Mod(0,2)); for(i=1,#v, if(s,v[i]=2-v[i]); s+=v[i]); fromdigits(v,9)<<2; \\ Kevin Ryde, Nov 06 2020

a(n) = 4*A163344(n).

a(n) = A163332(A338086(n)) = A338086(A128173(n)). - Kevin Ryde, Nov 06 2020

A165464 Squared distance between n's location in A163334 array and A163357 array.

Original entry on oeis.org

0, 0, 2, 4, 2, 4, 2, 0, 0, 2, 2, 4, 4, 4, 2, 0, 0, 2, 2, 4, 4, 2, 0, 0, 0, 0, 2, 4, 4, 2, 8, 10, 16, 16, 4, 2, 2, 10, 16, 10, 8, 8, 20, 20, 20, 18, 18, 32, 18, 10, 4, 2, 4, 10, 8, 2, 2, 10, 10, 4, 4, 4, 2, 10, 16, 26, 20, 10, 2, 4, 10, 18, 32, 32, 50, 52, 52, 34, 40, 58, 80, 80, 106, 146

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

Equivalently, squared distance between n's location in A163336 array and A163359 array. See example at A166041.

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

Positions of zeros: A165465. See also A165466, A163897, A163900.

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

A163344 Central diagonal of A163334 and A163336 divided by 4.

Original entry on oeis.org

0, 1, 2, 11, 10, 9, 18, 19, 20, 101, 100, 99, 90, 91, 92, 83, 82, 81, 162, 163, 164, 173, 172, 171, 180, 181, 182, 911, 910, 909, 900, 901, 902, 893, 892, 891, 810, 811, 812, 821, 820, 819, 828, 829, 830, 749, 748, 747, 738, 739, 740, 731, 730, 729, 1458

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

a(n) = A163343(n)/4.

A163479 Row sums of A163334 and A163336 divided by 6.

Original entry on oeis.org

0, 1, 2, 12, 22, 31, 40, 50, 60, 151, 242, 332, 422, 513, 604, 686, 768, 849, 930, 1012, 1094, 1185, 1276, 1366, 1456, 1547, 1638, 2458, 3278, 4097, 4916, 5736, 6556, 7367, 8178, 8988, 9798, 10609, 11420, 12240, 13060, 13879, 14698, 15518, 16338

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

a(n) = floor(A163342(n)/6) (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

A054238 Array read by downward antidiagonals: T(i,j) = bits of binary expansion of i interleaved with that of j.

Original entry on oeis.org

0, 1, 2, 4, 3, 8, 5, 6, 9, 10, 16, 7, 12, 11, 32, 17, 18, 13, 14, 33, 34, 20, 19, 24, 15, 36, 35, 40, 21, 22, 25, 26, 37, 38, 41, 42, 64, 23, 28, 27, 48, 39, 44, 43, 128, 65, 66, 29, 30, 49, 50, 45, 46, 129, 130, 68, 67, 72, 31, 52, 51, 56, 47, 132, 131, 136, 69, 70, 73, 74

Offset: 0

Marc LeBrun, Feb 07 2000

Inverse of sequence A054239 considered as a permutation of the nonnegative integers.
Permutation of nonnegative integers. Can be used as natural alternate number casting for pairs/tables (vs. usual diagonalization).
This array is a Z-order curve in an N x N grid. - Max Barrentine, Sep 24 2015
Each row n of this array is the lexicographically earliest sequence such that no term occurs in a previous row, no three terms form an arithmetic progression, and the k-th term in the n-th row is equal to the k-th term in row 0 plus some constant (specifically, T(n,k) = T(0,k) + A062880(n)). - Max Barrentine, Jul 20 2016

			From _Philippe Deléham_, Oct 18 2011: (Start)
The array starts in row n=0 with columns k >= 0 as follows:
   0  1  4  5 16 17 20 21 ...
   2  3  6  7 18 19 22 23 ...
   8  9 12 13 24 25 28 29 ...
  10 11 14 15 26 27 30 31 ...
  32 33 36 37 48 49 52 53 ...
  34 35 38 39 50 51 54 55 ...
  40 41 44 45 56 57 60 61 ...
  42 43 46 47 58 59 62 63 ...
(End)
T(6,5)=57 because 1.1.0. (6) merged with .1.0.1 (5) is 111001 (57). [Corrected by _Georg Fischer_, Jan 21 2022]

Robert Israel, Table of n, a(n) for n = 0..10000
G. M. Morton, A Computer Oriented Geodetic Data Base; and a New Technique in File Sequencing, IBM, 1966.
Wikipedia, Z-order Curve
Index entries for sequences that are permutations of the natural numbers

Cf. A000695 (row n=0), A062880 (column k=0), A001196 (main diagonal).
Cf. A059905, A059906, A346453 (by upwards antidiagonals).
See also A163357 and A163334 for other fractal curves in N x N grids.

N:= 4: # to get the first 2^(2N+1)+2^N terms
G:= 1/(1-y)/(1-x)*(add(2^(2*i+1)*x^(2^i)/(1+x^(2^i)),i=0..N) + add(2^(2*i)*y^(2^i)/(1+y^(2^i)),i=0..N)):
S:= mtaylor(G,[x=0,y=0],2^(N+1)):
seq(seq(coeff(coeff(S,x,i),y,m-i),i=0..m),m=0..2^(N+1)-1); # Robert Israel, Jul 21 2016

Table[Total@ Map[FromDigits[#, 2] &, Insert[#, 0, {-1, -1}] &@ Map[Riffle[IntegerDigits[#, 2], 0, 2] &, {n - k, k}]], {n, 0, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 21 2016 *)

T(n,k) = A000695(k) + 2*A000695(n). - Philippe Deléham, Oct 18 2011
From Robert Israel, Jul 21 2016: (Start)
G.f. of array: g(x,y) = (1/(1-x)*(1-y)) * Sum_{i>=0}
(2^(2*i+1)*x^(2^i)/(1+x^(2^i)) + 2^(2*i)*y^(2^i)/(1+y^(2^i))).
T(2*n+i,2*k+j) = 4*T(n,k) + 2*i+j for i,j in {0,1}. (End)

cp's OEIS Frontend

A163481 Row 0 of A163336 (column 0 of A163334).

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

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A166042 Permutation of nonnegative integers: a(n) tells which integer is in the same position in the square array A163334 as where n is located in the array A163357.

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A163342 Row sums of A163334 and A163336.

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A163343 Central diagonal of A163334 and A163336.

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A165464 Squared distance between n's location in A163334 array and A163357 array.

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A163344 Central diagonal of A163334 and A163336 divided by 4.

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A163479 Row sums of A163334 and A163336 divided by 6.

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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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A054238 Array read by downward antidiagonals: T(i,j) = bits of binary expansion of i interleaved with that of j.

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