A163899 -id:A163899 - cp's OEIS frontend

A165403 The positions of zeros in A163898 and A163899.

0, 1, 10, 21, 28, 136, 171, 190, 300, 325, 406, 465, 496, 2080, 2211, 2278, 2628, 2701, 2926, 3081, 3160, 4656, 4753, 5050, 5253, 5356, 6328, 6555, 6670, 7260, 7381, 7750, 8001, 8128, 32896, 33411, 33670, 34980, 35245, 36046, 36585, 36856

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

Equivalently, the positions of ones in A163904.

These positions are all in the top row of the said arrays, that is, for all n, A002262(a(n)) = 0. A025581(a(n)) gives A165404.

A165404 The positions of zeros in the top row of A163898 (and A163899).

Original entry on oeis.org

0, 1, 4, 6, 7, 16, 18, 19, 24, 25, 28, 30, 31, 64, 66, 67, 72, 73, 76, 78, 79, 96, 97, 100, 102, 103, 112, 114, 115, 120, 121, 124, 126, 127, 256, 258, 259, 264, 265, 268, 270, 271, 288, 289, 292, 294, 295, 304, 306, 307, 312, 313, 316, 318, 319, 384, 385, 388

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

Equivalently, the positions of ones in the top row of A163904.

A. Karttunen, Table of n, a(n) for n = 0..609

a(n) = A025581(A165403(n)). Same sequence in binary: A165406. For n>0, A147600(n-1) seems to give the number of terms with binary width n. See also A163901.

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

A163904 Array A(i,j): Cycle size of each A054238(i,j) in permutation A163355, 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

1, 1, 2, 6, 2, 6, 3, 3, 6, 3, 1, 6, 6, 3, 4, 3, 3, 3, 6, 8, 8, 1, 3, 4, 3, 4, 4, 2, 1, 2, 2, 2, 2, 2, 2, 2, 6, 2, 4, 4, 4, 4, 8, 2, 6, 6, 6, 2, 4, 4, 8, 8, 4, 6, 6, 12, 6, 12, 2, 4, 8, 4, 4, 12, 6, 18, 6, 6, 12, 6, 2, 4, 8, 4, 6, 12, 18, 6, 3, 12, 18, 6, 9, 2, 2, 8, 6, 6, 12, 18, 3, 3, 6, 18, 18

Offset: 0

Antti Karttunen, Sep 19 2009

			The top left 8x8 corner of this array:
1 1 6 3 1 3 1 1
2 2 3 6 3 3 2 2
6 6 6 3 4 2 4 2
3 3 6 3 2 4 4 2
4 8 4 2 4 4 4 2
8 4 2 4 8 8 4 2
2 2 8 8 4 8 2 2
2 2 4 4 4 8 2 2

A. Karttunen, Table of n, a(n) for n = 0..32895

a(n) = A163890(A054238(n)) = A163890(A163357(n)). Positions of 1's: A165403. See also A163898, A163899.

A163898 Array A(i,j) giving the square of distance from (i,j) to the location where A054238(i,j) is situated in array A163357(i,j), 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, 0, 1, 8, 1, 4, 18, 5, 2, 9, 0, 5, 2, 5, 16, 2, 1, 2, 9, 10, 25, 0, 1, 4, 9, 16, 17, 36, 0, 1, 4, 9, 16, 25, 36, 49, 128, 1, 2, 1, 10, 9, 10, 25, 64, 162, 113, 4, 5, 18, 5, 4, 5, 50, 81, 128, 113, 100, 9, 10, 5, 10, 1, 64, 65, 100, 128, 113, 100, 89, 16, 17, 20, 25, 64, 81, 100

Offset: 0

Antti Karttunen, Sep 19 2009

			The top left 8 X 8 corner of this array:
   0  0  8 18  0  2  0  0
   1  1  5  5  1  1  1  1
   4  2  2  2  4  4  2  4
   9  5  9  9  9  1  5  9
  16 10 16 16 10 18 10 16
  25 17 25  9  5  5 17 25
  36 36 10  4 10 20 36 36
  49 25  5  1 25 29 25 49

A. Karttunen, Table of n, a(n) for n = 0..32895

a(n) = A163900(A054238(n)). Positions of zeros: A165403. See also A163899, A163904.

A163900 Squared distance between n's location in A054238 array and A163357 array.

Original entry on oeis.org

0, 0, 1, 1, 8, 18, 5, 5, 4, 2, 9, 5, 2, 2, 9, 9, 0, 2, 1, 1, 0, 0, 1, 1, 4, 4, 9, 1, 2, 4, 5, 9, 16, 10, 25, 17, 16, 16, 25, 9, 36, 36, 49, 25, 10, 4, 5, 1, 10, 18, 5, 5, 10, 16, 17, 25, 10, 20, 25, 29, 36, 36, 25, 49, 128, 162, 113, 113, 128, 128, 113, 145, 100, 100, 89, 113, 162

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

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

Positions of zeros: A163901. See also A163898, A163899.

a(n) = A000290(abs(A059906(n)-A059252(n))) + A000290(abs(A059905(n)-A059253(n))).

cp's OEIS Frontend

A165403 The positions of zeros in A163898 and A163899.

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A165404 The positions of zeros in the top row of A163898 (and A163899).

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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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A163904 Array A(i,j): Cycle size of each A054238(i,j) in permutation A163355, 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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A163898 Array A(i,j) giving the square of distance from (i,j) to the location where A054238(i,j) is situated in array A163357(i,j), 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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A163900 Squared distance between n's location in A054238 array and A163357 array.

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