A163355 -id:A163355 - cp's OEIS frontend

A163914 Number of 3-cycles in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

Original entry on oeis.org

0, 0, 2, 1, 10, 9, 54, 57, 295, 329, 1613, 1834, 8812, 10072

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

a(n) = A163913(n)/3. Bisections: A163909, A163919. See also A163903, A163911, A163912, A163904, A163890.

A302845 Permutation of nonnegative integers: a(n) = A163355(A064707(n)).

Original entry on oeis.org

0, 1, 3, 2, 15, 14, 12, 13, 5, 6, 4, 7, 10, 9, 11, 8, 21, 20, 22, 23, 16, 19, 17, 18, 26, 27, 25, 24, 31, 28, 30, 29, 63, 62, 60, 61, 48, 49, 51, 50, 58, 57, 59, 56, 53, 54, 52, 55, 42, 43, 41, 40, 47, 44, 46, 45, 37, 36, 38, 39, 32, 35, 33, 34, 255, 254, 252, 253, 240, 241, 243, 242, 250, 249, 251, 248, 245, 246, 244

Offset: 0

Antti Karttunen, Apr 14 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences that are permutations of the natural numbers

Cf. A302846 (inverse).

Cf. A006068, A057300, A064707, A163355, A302843, A302782.

A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
A064707(n) = A006068(A006068(n));
A057300(n) = { my(t=1,s=0); while(n>0, if(1==(n%4),n++,if(2==(n%4),n--)); s += (n%4)*t; n >>= 2; t <<= 2); (s); };
A163355(n) = if(!n,n,my(i = (#binary(n)-1)\2, f = 4^i, d = (n\f)%4, r = (n%f)); if(((1==d)&&!(i%2))||((2==d)&&(i%2)), f+A163355(A057300(r)), if(3==d,f+f+A163355(A057300(r)), (3*f)+A163355(f-1-r))));
A302845(n) = A163355(A064707(n));

a(n) = A163355(A064707(n)).

a(n) = A302843(A006068(n)).

A163903 The positions i where A163915(i) = i, but not A163355(i) = i, that is, the 3-cycles of permutation A163355.

Original entry on oeis.org

5, 6, 10, 11, 13, 15, 17, 18, 19, 80, 81, 85, 86, 90, 91, 93, 95, 105, 106, 160, 161, 165, 166, 170, 171, 173, 175, 190, 191, 213, 215, 240, 241, 245, 246, 250, 251, 253, 255, 257, 258, 259, 276, 277, 278, 279, 282, 296, 297, 298, 299, 303, 309, 316, 317, 318

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

See also A163901, A163902, A163910, A163913, A163914.

A302843 Permutation of nonnegative integers: a(n) = A163355(A006068(n)).

Original entry on oeis.org

0, 1, 2, 3, 12, 13, 14, 15, 10, 9, 8, 11, 4, 7, 6, 5, 26, 27, 24, 25, 30, 29, 28, 31, 16, 19, 18, 17, 22, 23, 20, 21, 42, 43, 40, 41, 46, 45, 44, 47, 32, 35, 34, 33, 38, 39, 36, 37, 58, 57, 56, 59, 52, 55, 54, 53, 48, 49, 50, 51, 60, 61, 62, 63, 192, 193, 194, 195, 204, 205, 206, 207, 202, 201, 200, 203, 196, 199, 198, 197, 218, 219, 216, 217, 222

Offset: 0

Antti Karttunen, Apr 14 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences that are permutations of the natural numbers

Cf. A302844 (inverse).

Cf. A003188, A006068, A057300, A163355, A302845.

A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
A057300(n) = { my(t=1, s=0); while(n>0,  if(1==(n%4),n++,if(2==(n%4),n--)); s += (n%4)*t; n >>= 2; t <<= 2); (s); };
A163355(n) = if(!n,n,my(i = (#binary(n)-1)\2, f = 4^i, d = (n\f)%4, r = (n%f)); if(((1==d)&&!(i%2))||((2==d)&&(i%2)), f+A163355(A057300(r)), if(3==d,f+f+A163355(A057300(r)), (3*f)+A163355(f-1-r))));
A302843(n) = A163355(A006068(n));

a(n) = A163355(A006068(n)).

a(n) = A302845(A003188(n)).

A163902 The positions i where A163905(i) = i, but not A163355(i) = i, that is, the 2-cycles of permutation A163355.

Original entry on oeis.org

2, 3, 22, 23, 25, 26, 29, 31, 37, 38, 40, 41, 42, 43, 53, 55, 60, 61, 62, 63, 262, 263, 265, 266, 269, 271, 322, 323, 342, 343, 345, 346, 349, 351, 357, 358, 360, 361, 362, 363, 373, 375, 380, 381, 382, 383, 405, 406, 408, 409, 410, 411, 416, 420, 421, 422

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

A163913 Number of integers i in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356 with A163915(i)=i, but not A163355(i)=i.

Original entry on oeis.org

0, 0, 6, 3, 30, 27, 162, 171, 885, 987, 4839, 5502, 26436, 30216

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

a(n) = 3*A163914(n). See also A163903.

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

A163332 Self-inverse permutation of the integers for constructing the Peano curve in an N X N grid.

Original entry on oeis.org

0, 1, 2, 5, 4, 3, 6, 7, 8, 15, 16, 17, 14, 13, 12, 9, 10, 11, 18, 19, 20, 23, 22, 21, 24, 25, 26, 47, 46, 45, 48, 49, 50, 53, 52, 51, 44, 43, 42, 39, 40, 41, 38, 37, 36, 29, 28, 27, 30, 31, 32, 35, 34, 33, 54, 55, 56, 59, 58, 57, 60, 61, 62, 69, 70, 71, 68, 67, 66, 63, 64, 65

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

The integers [0,(3^k)-1] are confined to range [0,(3^k)-1].
From Kevin Ryde, Sep 04 2020: (Start)
To calculate a(n), write n in ternary digits n[k],..,n[0], where n[0] is the least significant digit.  Then the ternary digits of a(n) are a[j] = k^{n[j+1]+n[j+3]+...}(n[j]) where Peano's complement operator k^{s}(d) = d if s even or 2-d if s odd.
A single complement is k(d) = 2-d and the "exponent" is repeats k(k(k(...))).  Sum s = n[j+1] + n[j+3] + ... is every second digit above j, so digit j flips 0 <-> 2 when an odd number of odd digits (1's) among these.  The complement does not change digit parity so a second transformation re-complements back to the original digits and so self-inverse a(a(n)) = n.
Peano's curve is formed by digits of a(n) put alternately to x and y coordinates, so a(n) maps between the Peano curve the ternary Z-order curve per the formulas in A163528, A163529.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..59048
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).
Index entries for sequences that are permutations of the natural numbers

Coordinates using this transform: A163528, A163529.
A163334 & A163336 give two variants of the Peano curve in an N X N grid.
Cf. A163355 (Hilbert curve).

a[n_] := FromDigits[With[{d = Reverse@IntegerDigits[n, 3]}, Reverse@Table[
  If[EvenQ@Total@d[[j+1 ;; ;; 2]], d[[j]], 2-d[[j]]], {j, Length@d}]], 3];
Array[a, 100] (* Andrey Zabolotskiy, Apr 08 2021, after Kevin Ryde *)

a(n) = my(v=digits(n,3)); for(start=2,3, my(s=0); forstep(i=start,#v,2, s+=v[i-1]; if(s%2,v[i]=2-v[i]))); fromdigits(v,3); \\ Kevin Ryde, Sep 04 2020

a(n) = A163327(A163333(A163327(n))).

Name corrected by Kevin Ryde, Aug 27 2020

A163485 Permutation of integers used for constructing A147995 and A163545.

Original entry on oeis.org

0, 3, 1, 2, 14, 15, 13, 12, 6, 7, 5, 4, 8, 11, 9, 10, 58, 57, 59, 56, 62, 63, 61, 60, 54, 55, 53, 52, 50, 49, 51, 48, 26, 25, 27, 24, 30, 31, 29, 28, 22, 23, 21, 20, 18, 17, 19, 16, 32, 35, 33, 34, 46, 47, 45, 44, 38, 39, 37, 36, 40, 43, 41, 42, 234, 233, 235, 232, 228, 229

Offset: 0

Antti Karttunen, Aug 01 2009

nonn

Inverse: A163486. This permutation can be used to construct array A147995 and its transpose A163545. See A163355 for a bit similarly defined recursive permutation.

A163907 Permutation A163905 shown in N x N grid.

Original entry on oeis.org

0, 1, 2, 9, 3, 14, 10, 11, 12, 15, 16, 8, 4, 13, 47, 18, 19, 6, 7, 45, 44, 20, 17, 27, 5, 39, 46, 40, 21, 22, 25, 26, 37, 38, 41, 42, 149, 23, 30, 24, 58, 36, 33, 43, 234, 150, 151, 29, 28, 56, 57, 34, 35, 232, 235, 152, 148, 157, 31, 54, 59, 49, 32, 236, 233, 227, 154

Offset: 0

Antti Karttunen, Sep 19 2009

			The top left 8x8 corner of this array:
+0 +1 +9 10 16 18 20 21
+2 +3 11 +8 19 17 22 23
14 12 +4 +6 27 25 30 29
15 13 +7 +5 26 24 28 31
47 45 39 37 58 56 54 53
44 46 38 36 57 59 52 55
40 41 33 34 49 51 60 61
42 43 35 32 48 50 62 63

Inverse: A163908. a(n) = A163905(A054238(n)) = A163355(A163357(n)). See also A163357, A163917.

cp's OEIS Frontend

A163914 Number of 3-cycles in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356.

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A302845 Permutation of nonnegative integers: a(n) = A163355(A064707(n)).

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A163903 The positions i where A163915(i) = i, but not A163355(i) = i, that is, the 3-cycles of permutation A163355.

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A302843 Permutation of nonnegative integers: a(n) = A163355(A006068(n)).

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A163902 The positions i where A163905(i) = i, but not A163355(i) = i, that is, the 2-cycles of permutation A163355.

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A163913 Number of integers i in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356 with A163915(i)=i, but not A163355(i)=i.

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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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A163332 Self-inverse permutation of the integers for constructing the Peano curve in an N X N grid.

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A163485 Permutation of integers used for constructing A147995 and A163545.

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A163907 Permutation A163905 shown in N x N grid.

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