A163328 -id:A163328 - cp's OEIS frontend

A163329 Inverse permutation to A163328.

0, 1, 3, 2, 4, 7, 5, 8, 12, 6, 10, 15, 11, 16, 22, 17, 23, 30, 21, 28, 36, 29, 37, 46, 38, 47, 57, 9, 13, 18, 14, 19, 25, 20, 26, 33, 24, 31, 39, 32, 40, 49, 41, 50, 60, 48, 58, 69, 59, 70, 82, 71, 83, 96, 27, 34, 42, 35, 43, 52, 44, 53, 63, 51, 61, 72, 62, 73, 85, 74, 86, 99

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

Inverse: A163328. a(n) = A163331(A163327(n)). A054239 is an analogous sequence for binary. Cf. A007089.

(define (A163329 n) (A001477bi (A163325 n) (A163326 n)))
(define (A001477bi x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))

a(n) = A001477bi(A163325(n),A163326(n)), where A001477bi(x,y) = (((x+y)^2)+x+(3y))/2.

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

A163336 Peano curve in an n X n grid, starting downwards 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, 5, 1, 6, 4, 2, 47, 7, 3, 15, 48, 46, 8, 14, 16, 53, 49, 45, 9, 13, 17, 54, 52, 50, 44, 10, 12, 18, 59, 55, 51, 39, 43, 11, 23, 19, 60, 58, 56, 38, 40, 42, 24, 22, 20, 425, 61, 57, 69, 37, 41, 29, 25, 21, 141, 426, 424, 62, 68, 70, 36, 30, 28, 26, 140, 142, 431, 427

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  5  6 47 48 53 54 59 60
   1  4  7 46 49 52 55 58 61
   2  3  8 45 50 51 56 57 62
  15 14  9 44 39 38 69 68 63
  16 13 10 43 40 37 70 67 64
  17 12 11 42 41 36 71 66 65
  18 23 24 29 30 35 72 77 78
  19 22 25 28 31 34 73 76 79
  20 21 26 27 32 33 74 75 80

A. Karttunen, Table of n, a(n) for n = 0..3320
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 (figure 3 with Y downwards is 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).
Rémy Sigrist, Colored scatterplot of (x, y) such that 0 <= x, y < 3^6 (where the hue is function of T(x, y))
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: A163334. Inverse: A163337. a(n) = A163332(A163330(n)) = A163327(A163333(A163328(n))) = A163334(A061579(n)). One-based version: A163340. Row sums: A163342. Row 0: A163481. Column 0: A163480. Central diagonal: A163343.

See A163357 and A163359 for the Hilbert curve.

b[{n_, k_}, {m_}] := (A[n, k] = 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 *)

Name corrected by Kevin Ryde, Aug 28 2020

A163327 Self-inverse permutation of integers: swap the odd- and even-positioned digits in the ternary expansion of n, then convert back to decimal.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jul 29 2009

			11 in ternary base (A007089) is written as '(000...)102' (... + 0*27 + 1*9 + 0*3 + 2), which results '1020' = 1*27 + 0*9 + 2*3 + 0 = 33, when the odd- and even-positioned digits are swapped, thus a(11) = 33.

Antti Karttunen, Table of n, a(n) for n = 0..728
Index entries for sequences that are permutations of the natural numbers

Cf. A007089, A163328-A163329.

Other bases: A057300, A126006, A217558.

from sympy.ntheory import digits
def a(n):
    d = digits(n, 3)[1:]
    return sum(3**(i+(1-2*(i&1)))*di for i, di in enumerate(d[::-1]))
print([a(n) for n in range(72)]) # Michael S. Branicky, Aug 05 2022

(define (A163327 n) (+ (A037314 (A163326 n)) (* 3 (A037314 (A163325 n)))))

a(n) = A037314(A163326(n)) + 3*A037314(A163325(n))

Edited by Charles R Greathouse IV, Nov 01 2009

A163325 Pick digits at the even distance from the least significant end of the ternary expansion of n, then convert back to decimal.

Original entry on oeis.org

0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, 6, 7, 8, 6, 7, 8, 6, 7, 8, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, 6, 7, 8, 6, 7, 8, 6, 7, 8, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, 6, 7, 8, 6, 7, 8, 6, 7, 8, 9, 10, 11, 9, 10, 11, 9, 10, 11, 12, 13, 14, 12, 13, 14

Offset: 0

Antti Karttunen, Jul 29 2009

			11 in ternary base (A007089) is written as '102' (1*9 + 0*3 + 2), from which we pick the "zeroth" and 2nd digits from the right, giving '12' = 1*3 + 2 = 5, thus a(11) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..728
Kevin Ryde, Plot2 of X=A163325,Y=A163326, illustrating the ternary Z-order curve.
Index entries for sequences related to coordinates of 2D curves

A059905 is an analogous sequence for binary.

Cf. A007089, A163327, A163328, A163329.

a(n) = fromdigits(digits(n,9)%3,3); \\ Kevin Ryde, May 14 2020

a(0) = 0, a(n) = (n mod 3) + 3*a(floor(n/9)).
a(n) = Sum_{k>=0} {A030341(n,k)*b(k)} where b is the sequence (1,0,3,0,9,0,27,0,81,0,243,0... = A254006): powers of 3 alternating with zeros. - Philippe Deléham, Oct 22 2011
A037314(a(n)) + 3*A037314(A163326(n)) = n for all n.

Edited by Charles R Greathouse IV, Nov 01 2009

A338086 Duplicate the ternary digits of n, so each 0, 1 or 2 becomes 00, 11 or 22 respectively.

Original entry on oeis.org

0, 4, 8, 36, 40, 44, 72, 76, 80, 324, 328, 332, 360, 364, 368, 396, 400, 404, 648, 652, 656, 684, 688, 692, 720, 724, 728, 2916, 2920, 2924, 2952, 2956, 2960, 2988, 2992, 2996, 3240, 3244, 3248, 3276, 3280, 3284, 3312, 3316, 3320, 3564, 3568, 3572, 3600, 3604

Offset: 0

Kevin Ryde, Oct 09 2020

Also, numbers whose ternary digit runs are all even lengths (including 0 reckoned as no digits at all).  Also, change ternary digits 0,1,2 to base 9 digits 0,4,8, and hence numbers which can be written in base 9 using only digits 0,4,8.
Digit duplication 00,11,22 can be compared to A037314 which is 0 above each so 00,01,02, or A208665 which is 0 below each so 00,10,20.  Duplication is the sum of these, or any one is a suitable multiple of another (*3, *4, etc).
This sequence is the points on the X=Y diagonal of the ternary Z-order curve (see example table in A163328).  The Z-order curve takes a point number p and splits its ternary digits alternately to X and Y coordinates so X(p) = A163325(p) and Y(p) = A163326(p).  Duplicate digits in a(n) are the diagonal X(a(n)) = Y(a(n)) = n.

			n=73 is ternary 2201 which duplicates to 22220011 ternary = 8804 base 9 = 6484 decimal.

Kevin Ryde, Table of n, a(n) for n = 0..6560

Cf. A037314, A208665, A163325, A163326, A163328.
Cf. A020331 (ternary concatenation).
Digit duplication in other bases: A001196, A338754.

PARI
```
a(n) = fromdigits(digits(n,3),9)<<2;
```

from gmpy2 import digits
def A338086(n): return int(''.join(d*2 for d in digits(n,3)),3) # Chai Wah Wu, May 07 2022

a(n) = A037314(n) + A208665(n) = 4*A037314(n) = (4/3)*A208665(n).

a(n) = 4*Sum_{i=0..k} d[i]*9^i where the ternary expansion of n is n = Sum_{i=0..k} d[i]*3^i with digits d[i]=0,1,2.

A163330 Square array A, where entry A(y,x) has the ternary digits of y interleaved with the ternary digits of x, converted back to decimal. Listed by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

Original entry on oeis.org

0, 3, 1, 6, 4, 2, 27, 7, 5, 9, 30, 28, 8, 12, 10, 33, 31, 29, 15, 13, 11, 54, 34, 32, 36, 16, 14, 18, 57, 55, 35, 39, 37, 17, 21, 19, 60, 58, 56, 42, 40, 38, 24, 22, 20, 243, 61, 59, 63, 43, 41, 45, 25, 23, 81, 246, 244, 62, 66, 64, 44, 48, 46, 26, 84, 82, 249, 247, 245

Offset: 0

Antti Karttunen, Jul 29 2009

Inverse: A163331. a(n) = A163327(A163328(n)). Transpose: A163328. Cf. A007089, A163327, A163332, A163334.

(define (A163330 n) (+ (A037314 (A002262 n)) (* 3 (A037314 (A025581 n)))))

a(n) = 3*A037314(A025581(n)) + A037314(A002262(n))

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A163329 Inverse permutation to A163328.

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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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A163336 Peano curve in an n X n grid, starting downwards 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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A163327 Self-inverse permutation of integers: swap the odd- and even-positioned digits in the ternary expansion of n, then convert back to decimal.

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A163325 Pick digits at the even distance from the least significant end of the ternary expansion of n, then convert back to decimal.

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A338086 Duplicate the ternary digits of n, so each 0, 1 or 2 becomes 00, 11 or 22 respectively.

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A163330 Square array A, where entry A(y,x) has the ternary digits of y interleaved with the ternary digits of x, converted back to decimal. Listed by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

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