A163334 -id:A163334 - cp's OEIS frontend

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), ...

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

A163359 Hilbert curve in N x N grid, starting downwards from the top-left corner, listed by descending antidiagonals.

Original entry on oeis.org

0, 3, 1, 4, 2, 14, 5, 7, 13, 15, 58, 6, 8, 12, 16, 59, 57, 9, 11, 17, 19, 60, 56, 54, 10, 30, 18, 20, 63, 61, 55, 53, 31, 29, 23, 21, 64, 62, 50, 52, 32, 28, 24, 22, 234, 65, 67, 49, 51, 33, 35, 27, 25, 233, 235, 78, 66, 68, 48, 46, 34, 36, 26, 230, 232, 236, 79, 77, 71

Offset: 0

Antti Karttunen, Jul 29 2009

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

A. Karttunen, Table of n, a(n) for n = 0..32895
David Hilbert, Ueber die stetige Abbildung einer Linie auf ein Flächenstück, Mathematische Annalen, volume 38, number 3, 1891, pages 459-460. Also EUDML (link to GDZ).
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: A163357, a(n) = A163357(A061579(n)). Inverse: A163360. One-based version: A163363. Row sums: A163365. Row 0: A163483. Column 0: A163482. Central diagonal: A062880.

See also A163334 and A163336 for the Peano curve.

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

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

A147995 Array of N X N grid hopping "almost-walk", read by antidiagonals.

Original entry on oeis.org

0, 1, 3, 6, 2, 14, 5, 7, 13, 15, 26, 4, 8, 12, 58, 27, 25, 9, 11, 59, 57, 22, 24, 30, 10, 54, 56, 62, 21, 23, 29, 31, 53, 55, 61, 63, 106, 20, 18, 28, 32, 52, 50, 60, 234, 107, 105, 19, 17, 33, 35, 51, 49, 235, 233, 108, 104, 100, 16, 38, 34, 46, 48, 236, 232, 228, 111

Offset: 0

Roger L. Bagula and Gary W. Adamson, Nov 18 2008

The original name was: "The sequence is an anti-diagonal of the decimal of a mapped 4-ary Gray code matrix as a triangular sequence."
Gary W. Adamson's explanation of the sequence: Here's the conversion rules for the codons, 4-Ary gray code, which "turns out" to be the most appropriate format for mapping the Codons on a gray code Karnaugh map. The "why" this is the appropriate format relates to a degree of trial and error to find the proper fit in terms of the numbers of hydrogen bonds per codon- anticodon. (Antti Karttunen's comment: obscure definition. The "degree of trial and error" should be defined transparently.)
1) The "H-bond codon-anticodon magic square" map by Gary Adamson, published on page 287 of Cliff Pickover's book "Zen of Magic Squares..." looks like this:
  CCC CCU CUU CUC UUC UUU UCU UCC
  CCA CCG CUG CUA UUA UUG UCG UCA
  CAA CAG CGG CGA UGA UGG UAG UAA
  CAC CAU CGU CGC UGC UGU UAU UAC
  AAC AAU AGU AGC GGC GGU GAU GAC
  AAA AAG AGG AGA GGA GGG GAG GAA
  ACA ACG AUG AUA GUA GUG GCG GCA
  ACC ACU AUU AUC GUC GUU GCU GCC
2) Using the conversion rules: 0 = C, 1 = A, 2 = G, 3 = U, we convert to 4-ary gray code:
  000 003 033 030 330 333 303 300
  001 002 032 031 331 332 302 301
  011 012 022 021 321 322 312 311
  010 013 023 020 320 323 313 310
  110 113 123 120 220 223 213 210
  111 112 122 121 221 222 212 211
  101 102 132 131 231 232 202 201
  100 103 133 130 230 233 203 200
3) To convert back to decimal:
   0  3 14 15 58 57 62 63
   1  2 13 12 59 56 61 60
   6  7  8 11 54 55 50 49
   5  4  9 10 53 52 51 48
  26 25 30 31 32 35 46 47
  27 24 29 28 33 34 45 44
  22 23 18 17 38 39 40 43
  21 20 19 16 37 36 41 42
... and that's it! Notice how the 1,2,3,... jumps around, somewhat like a Peano curve, from one 4-unit cell to the next.
Antti Karttunen's notes: The steps 1 & 2 are clear, but the step 3 would not produce the array given here, but instead the array A163239. Furthermore, in Pickover's book the conversion rules C=0, A=1, U=2, G=3 are used, in which case we get the array A163235. Also, the path taken by the terms does not form a continuous Peano curve (Hamiltonian path), because there are discontinuities, e.g., when going from 3 to 4, or from 15 to 16. See A163357/A163359 & A163334/A163336 for examples of continuous Peano/Hilbert curves/paths in an N X N grid. However, this sequence is uniquely defined by the formula a(n) = A163485(A057300(A054238(n))). The 8 X 8 array given at the step 3 is the top left corner of the infinite square array whose antidiagonal gives this sequence.
From Gary W. Adamson, Aug 04 2009: (Start)
This entry was originally only an e mail to the coauthor; but given that the terms are correct, the complete set of rules for the system can be presented.
Using 3 bit terms, we write out the Gray code for (0 - 7) as row headings; doing the same as the left column, then each of the 64 entries places the left column term (of 3 bits) underneath the top row headings. Then reading 2 bits from top to down in each entry, we use (0,0) = C; (1,1) = G; (0,1) = A and (1,0) = U. This gives the Gray code Karnaugh map along with 64 codons:
.
  000...001...011...010...110...111...101...100
  000...000...000...000...000...000...000...000
  CCC...CCU...CUU...CUC...UUC...UUU...UCU...UCC
  000...001...011...010...110...111...101...100
  001...001...001...001...001...001...001...001
  CCA...CCG...CUG...CUA...UUA...UUG...UCG...UCA
  000...001...011...010...110...111...101...100
  011...011...011...011...011...011...011...011
  CAA...CAG...CGG...CGA...UGA...UGG...UAG...UAA
  000...001...011...010...110...111...101...100
  010...010...010...010...010...010...010...010
  CAC...CAU...CGU...CGC...UGC...UGU...UAU...UAC
  000...001...011...010...110...111...101...100
  110...110...110...110...110...110...110...110
  AAC...AAU...AGU...AGC...GGC...GGU...GAU...GAC
  000...001...011...010...110...111...101...100
  111...111...111...111...111...111...111...111
  AAA...AAG...AGG...AGA...GGA...GGG...GAG...GAA
  000...001...011...010...110...111...101...100
  101...101...101...101...101...101...101...101
  ACA...ACG...AUG...AUA...GUA...GUG...GCG...GCA
  000...001...011...010...110...111...101...100
  100...100...100...100...100...100...100...100
  ACC...ACU...AUU...AUC...GUC...GUU...GCU...GCC
.
Next, reading again from top 3 bits to bottom, we convert the base-2 Gray code to 4-ary Gray code using the rules (0,0) = 0; (0,1) = 1; (1,1) = 2; and (1,0) = 3; giving the array given using numbers (0,1,2, and 3) = 4-ary Gray code. The previous 2 maps have the unique Gray code property of having only a 1 bit (or 1 letter) change in any direction: up, down, right, left, including wrap-arounds.
Last part of this system, we need create a linear system of Codons with only 1 bit (letter) change from one term to the next, giving an ordered decimal term for each Codon. This is done by converting the array with the (0,1,2,3) terms to the corresponding decimal term. Thus given the array: 000...003...033...030...330...333...etc; considered as 4-ary Gray code, these terms are equivalent to the array A147995 (then take antidiagonals).
Following the numbers in succession in the array (0 -> 1 -> 2 ->...63) allows for us to have a linear system of Codons with only a 1-letter change from one Codon to the next, as follows: CCC -> CCA -> CCG -> CAU...-> through 63 = UCC. The other entries as of this date in the OEIS do not have the 1-letter (only) change from one associated decimal term to the next. For example, take entry A163235: If the decimal number system (given) is superimposed upon the 64 Codon array, the term 3 corresponds to CCG, but 4 in the left column corresponds to CAC, having a 2-letter change. Similarly, take A163239: If the decimal array in that entry is superimposed on the 64 Codon array, "3" corresponds in position to CCU, but "4" corresponds to CAC; again a 2-letter change. The system given in A147995 preserves the unique 1 (bit/letter) change from one Codon to any neighbor, going in any direction; along with the corresponding linear system with a 1-letter change from one Codon to the next.
Last, we submit for each Codon the number of hydrogen bonds per codon/anti-codon using the following substitution rules: (C,G) = 3; (A,U) = 2, then add.
This gives following array which we superimpose on the Codon array, giving the correct number of Hydrogen bonds for each Codon and anti-Codon:
.
  9 8 7 8 7 6 7 8
  8 9 8 7 6 7 8 7
  7 8 9 8 7 8 7 6
  8 7 8 9 8 7 6 7
  7 6 7 8 9 8 7 8
  6 7 8 7 8 9 8 7
  6 8 7 6 7 8 9 8
  8 7 6 7 8 7 8 9
... (a semi-magic square with a binomial distribution of (1, 3, 3, 1) as to (6, 7, 8, 9) in every row and column.
Example: CUG (3rd from left, row next to top) has (C=3, U=2, G=3), total 8.
The anti-Codon of CUG = GAC and likewise has 8 hydrogen bonds. (End)
From Gary W. Adamson, Aug 04 2009: (Start)
The final outcome: superimposing the Codon map onto the decimal term map, we obtain a linear sequence of Codons with a 1-letter change between neighbors (which begs the question of how many such permutations are possible with the 1-letter change). The method of A147995 gives:
.
   0   CCC;     16   AUC;     32   GGC;     48   UAC
   1   CCA;     17   AUA;     33   GGA;     49   UAA
   2   CCG;     18   AUG;     34   GGG;     50   UAG
   3   CCU;     19   AUU;     35   GGU;     51   UAU
   4   CAU;     20   ACU;     36   GUU;     52   UGU
   5   CAC;     21   ACC;     37   GUC;     53   UGC
   6   CAA;     22   ACA;     38   GUA;     54   UGA
   7   CAG;     23   ACG;     39   GUG;     55   UGG
   8   CGG;     24   AAG;     40   GCG;     56   UUG
   9   CGU;     25   AAU;     41   GCU;     57   UUU
  10   CGC;     26   AAC;     42   GCC;     58   UUC
  11   CGA;     27   AAA;     43   GCA;     59   UUA
  12   CUA;     28   AGA;     44   GAA;     60   UCA
  13   CUG;     29   AGG;     45   GAG;     61   UCG
  14   CUU;     30   AGU;     46   GAU;     62   UCU
  15   CUC;     31   AGC;     47   GAC;     63   UCC
(End)
From Gary W. Adamson, Aug 08 2009: (Start)
The 8 X 8 array of hydrogen bonds can be derived from the 3rd row of A088696 (1, 2, 3, 2, 3, 4, 3, 2) using a simple conversion rule. Given the terms of A088696, each is replaced with its complement to 10: (1->9; 2->8; 3->7; 4->6) Note that the leftmost column going down should read: (9, 8, 7, 8, 7, 6, 7, 8) matching the top row from left to right. (End)
From Gary W. Adamson, Aug 13 2009: (Start)
Gray code -> <- Binary conversion rules: in either direction for any base; "N-Ary Gray code" -> "N-ary" or in the other direction.
.
First, N-Ary Gray code to N-Ary conversion. Write the N-Ary on a top row with the Gray code on the bottom row in both conversion variants. Given a Gray code on the bottom row, the N-Ary may be defined as "running sums MOD N" of the bottom row; then use the following rules: Leftmost term is the same.
Next, use the sum of term (n-th) in the top row from the left, and the (n+1)-th term in the bottom row, MOD N. By way of example:
Convert Gray code base 8, 3641063 to 8-ary. This gives initially,
3..................
3..6..4..1..0..6..3
.
Then (3 + 6) MOD 8 = 1 so we place a "1" above the 6 going to the right.
Then (1 + 4) MOD 8 = 5 so we place a "5" above the 5.
Continuing with this procedure, we obtain:
3 1 5 6 6 4 7 8-Ary
3 6 4 1 0 6 3 8-Ary Gray code
.
Using the 8 X 8 4-Ary chart, convert 133 (bottom row, 4th from the left) to 4-Ary then to decimal. Our setup is:
1
1 3 3
getting (1, 0, 3). Then placing powers of 4 above the 4-Ary, = 1*16 + 3 = 19 as shown in the accompanying chart, 4-Ary Gray code 133 = 19 decimal.
.
Rules for converting an N-Ary number to the corresponding N-Ary Gray code:
As before, we place the N-Ary on the top row with ongoing results on the bottom row = N-Ary Gray code.
In the top row from left to right, through through the entire number looking at pairs (n-th and (n+1)-th terms), if (n+1)-th is > than n-th, take the difference and write it down. If term (n+1) = n-th term, write down a "0".
If term (n+1) < n-th term we ADD N (as N-Ary) to (n+1)-th term then take the difference. Examples:
Find the Gray code counterpart to 2 1 base 4 = 9 decimal.
Ans.: next term (1) < (2) so we add 4 to the 1 getting 5, then take (5 - 2) = 3. So given 4-Ary 21, the corresponding Gray code term = 23
.
Find the Gray code counterpart to binary 10110 = 22 decimal. First, go through the terms writing down the difference if next term > current: (and writing "0" if next term = current term)
1, 0, 1, 1, 0
1.....1..0...
Add "2" to the terms above the vacant places and take the difference from previous term, top row:
1, 1, 1, 0, 1 final result = Gray code for 22 decimal.
.
Given 8-Ary number 3156647, base 8. Using steps (1-2) we get
3, 1, 5, 6, 6, 4, 7
3.....4..1..0.....3; then add 8 to top term for vacant places then take the difference, getting:
3..6..4..1..0..6..3; = 8-ary Gray code given 8-Ary (3 1 5 6 6 4 7).
.
Given the foregoing rules and examples, access the charts accompanying the DNA codons. 3 digit terms = 4-Ary Gray code. Convert 133 (bottom row) to 4-Ary then to decimal. We get:
1
1 0 3 = (16 + 0 + 3) = 19
Convert 39 decimal to 4-Ary then to 4-Ary Gray code. 39 = 213 4-Ary = (2*16 + 4 + 3); then
2 1 3
2...2; then add "4" to the 1 and take the difference = (5 - 2) = 3. = 2 3 2 = 4-Ary Gray code for decimal 39 as shown in the dual charts, next to bottom row, third from the right: (232 corresponds to 39) in the accompanying chart.
Properties of Gray code: sum of terms MOD N = decimal MOD N. Example: 232 corresponds to 19, then (2 + 3 + 2) MOD 4 = 3, and 19 == 3 MOD 4.
Another property: Highest exponent of N dividing a decimal term.
Access term (n-1) writing the Gray code on the top row and Gray code for n-th term on the bottom. Determine column change = (0, 1, 2, ...) starting from the right. Let the column = c. then c is the highest exponent of N dividing n-th term. Examples: 40 in 4-Ary Gray code = 202, while 41 = 203. Change is in column 0 so 203 can be divided by 4^0. But 44 in 4-Ary Gray code = 211 while 43 = 201. Bit change is in column 1 so 4^1 divides 44. (End)

			Antidiagonals begin:
  { 0},
  { 1,  3},
  { 6,  2, 14},
  { 5,  7, 13, 15},
  {26,  4,  8, 12, 58},
  {27, 25,  9, 11, 59, 57},
  {22, 24, 30, 10, 54, 56, 62},
  {21, 23, 29, 31, 53, 55, 61, 63}

Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions, Princeton University Press, 2002, pp. 285-289.

A. Karttunen, Table of n, a(n) for n = 0..8255
Jay Kappraff and Gary W. Adamson, Generalized Genomic Matrices, Silver Means, & Pythagorean Triples, FORMA 2009, v24 p.41-48.
Index entries for sequences that are permutations of the natural numbers

a(n) = A163545(A061579(n)), i.e., transpose of A163545. Antidiagonal sums: A163484. Inverse: A163544. See also A163233, A163235, A163237, A163239, A163357, A163359.

Cf. A088696. - Gary W. Adamson, Aug 08 2009

M = {{0, 3, 14, 15, 58, 57, 62, 63}, {1, 2, 13, 12, 59, 56, 61, 60}, {6, 7, 8, 11, 54, 55, 50, 49}, {5, 4, 9, 10, 53, 52, 51, 48}, {26, 25, 30, 31, 32, 35, 46, 47}, {27, 24, 29, 28, 33, 34, 45, 44}, {22, 23, 18, 17, 38, 39, 40, 43}, {21, 20, 19, 16, 37, 36, 41, 42}}; Table[Table[M[[n - m + 1, m]], {m, 1, n}], {n, 1, Length[M]}]; Flatten[%]

M = {{0, 3, 14, 15, 58, 57, 62, 63}, {1, 2, 13, 12, 59, 56, 61, 60}, {6, 7, 8, 11, 54, 55, 50, 49}, {5, 4, 9, 10, 53, 52, 51, 48}, {26, 25, 30, 31, 32, 35, 46, 47}, {27, 24, 29, 28, 33, 34, 45, 44}, {22, 23, 18, 17, 38, 39, 40, 43}, {21, 20, 19, 16, 37, 36, 41, 42}}; t(n,m) = antidiagonal(M).

a(n) = A163485(A057300(A054238(n))). - Antti Karttunen, Aug 01 2009

Edited, extended, keywords tabl and obsc added and offset changed from 1 to 0 by Antti Karttunen, Aug 01 2009

A163358 Inverse permutation to A163357.

Original entry on oeis.org

0, 1, 4, 2, 5, 9, 13, 8, 12, 18, 24, 17, 11, 7, 3, 6, 10, 16, 22, 15, 21, 28, 37, 29, 38, 47, 58, 48, 39, 30, 23, 31, 40, 50, 60, 49, 59, 70, 83, 71, 84, 97, 112, 98, 85, 72, 61, 73, 62, 52, 42, 51, 41, 32, 25, 33, 26, 19, 14, 20, 27, 34, 43, 35, 44, 54, 64, 53, 63, 74, 87

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

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

Inverse: A163357. a(n) = A054239(A163356(n)). One-based version: A163362. See also A163334 and A163336.

A163328 Square array A, where entry A(y,x) has the ternary digits of x interleaved with the ternary digits of y, 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, 1, 3, 2, 4, 6, 9, 5, 7, 27, 10, 12, 8, 28, 30, 11, 13, 15, 29, 31, 33, 18, 14, 16, 36, 32, 34, 54, 19, 21, 17, 37, 39, 35, 55, 57, 20, 22, 24, 38, 40, 42, 56, 58, 60, 81, 23, 25, 45, 41, 43, 63, 59, 61, 243, 82, 84, 26, 46, 48, 44, 64, 66, 62, 244, 246, 83, 85, 87, 47, 49

Offset: 0

Antti Karttunen, Jul 29 2009

			From _Kevin Ryde_, Oct 06 2020: (Start)
Array A(y,x) read by downwards antidiagonals, so 0, 1,3, 2,4,6, etc.
        x=0   1   2   3   4   5   6   7   8
      +--------------------------------------
  y=0 |   0,  1,  2,  9, 10, 11, 18, 19, 20,
    1 |   3,  4,  5, 12, 13, 14, 21, 22,
    2 |   6,  7,  8, 15, 16, 17, 24,
    3 |  27, 28, 29, 36, 37, 38,
    4 |  30, 31, 32, 39, 40,
    5 |  33, 34, 35, 42,
    6 |  54, 55, 56,
    7 |  57, 58,
    8 |  60,
(End)

Antti Karttunen, Table of n, a(n) for n = 0..3320
Index entries for sequences that are permutations of the natural numbers

Inverse: A163329. Transpose: A163330. Cf. A037314 (row y=0), A208665 (column x=0)

Cf. A054238 is an analogous sequence for binary. Cf. A007089, A163327, A163332, A163334.

A(y,x) = 3*fromdigits(digits(y,3),9) + fromdigits(digits(x,3),9); \\ Kevin Ryde, Oct 06 2020

(define (A163328 n) (+ (A037314 (A025581 n)) (* 3 (A037314 (A002262 n)))))

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

a(n) = A163327(A163330(n)).

Edited by Charles R Greathouse IV, Nov 01 2009

A057212 n-th run has length n.

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Ben Tyner (tyner(AT)phys.ufl.edu), Sep 27 2000

T(n,k) = 1 - n mod 2, 1 <= k <= n. [Reinhard Zumkeller, Mar 18 2011]

{a(n)} interpreted as a string over {0,1} is one of exactly two fixed-points of the function defined by f(0^n 1 s) = 1^(n-1) f(1 s) and f(1^n 0 s) = 0^(n-1) f(0 s). The other fixed point is obtained by swapping all 0s and 1s. - Curtis Bechtel, Jun 27 2025

K. H. Rosen, Discrete Mathematics and its Applications, 1999, fourth edition, page 79, exercise 10 (g).

Cf. A057211.

As a simple triangular or square array virtually the only sequences which appear are A000004, A000012 and A000035. Cf. A060510.

a057212 n = a057212_list !! (n-1)
a057212_list = concat $ zipWith ($) (map replicate [1..]) a000035_list
-- Reinhard Zumkeller, Mar 18 2011

A002024 := n->round(sqrt(2*n)):A057212 := n->(1+(-1)^A002024(n))/2;
# alternative Maple program:
T:= n-> [irem(1+n, 2)$n][]:
seq(T(n), n=1..14);  # Alois P. Heinz, Oct 06 2021

Table[If[OddQ[n], 0, 1], {n, 1, 14}, {n}] // Flatten (* Jean-François Alcover, Mar 07 2021 *)

from math import isqrt
def A057212(n): return int(not isqrt(n<<3)+1&2) # Chai Wah Wu, Jun 19 2024

a(n)=A003056(n) mod 2 so as a square array T(n, k)=n+k mod 2 - Henry Bottomley, Mar 22 2001
a(n) = (1+(-1)^A002024(n))/2, where A002024(n)=round(sqrt(2*n)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 23 2003
a(n)=A163334(n) mod 2 = A163336(n) mod 2 = A163357(n) mod 2 = A163359(n) mod 2, i.e. the array gives the parity of elements at the successive antidiagonals (alternating between 0 and 1) of square arrays constructed from ANY Hilbert curve starting from zero located at the top left corner of a square grid (and using only N,E,S,W steps of length one). - Antti Karttunen, Oct 22 2012
a(n) = 1 - A057211(n). - Alois P. Heinz, Oct 06 2021

A334188 T(n, k) is the number of steps from the point (0, 0) to the point (k, n) along the space filling curve U described in Comments section; a negative value corresponds to moving backwards; square array T(n, k), n, k >= 0 read by antidiagonals downwards.

Original entry on oeis.org

0, 1, -1, 2, -6, -2, 3, -7, -5, -3, 8, 4, -8, -4, -12, 9, 7, 5, -9, -11, -13, 10, 18, 6, -26, -10, -18, -14, 11, 17, 19, -27, -25, -19, -17, -15, 40, 12, 16, 20, -28, -24, -20, -16, -48, 41, 39, 13, 15, 21, -29, -23, -21, -47, -49, 42, 34, 38, 14, 22, -34, -30

Offset: 0

Rémy Sigrist, Apr 18 2020

We start with a unit square U_0 oriented counterclockwise, the origin being at the left bottom corner:
         +---<---+
         |       |
         v       ^
         |       |
         O--->---+
The configuration U_{k+1} is obtained by connecting four copies of the configuration U_k as follows:
             |   |                               |   |
         .   +   +   .                       .   +   +   .
     U_k     ^   v     U_k                       ^   v
         .   +   +   .                       .   +   +   .
             |   |                               |   |
    -+->-+---+   +---+->-+-             -+->-+   +   +   +->-+-
                                -->          v   |   |   ^
    -+-<-+---+   +---+-<-+-             -+-<-+   +-<-+   +-<-+-
             |   |
         .   +   +   .                       .   +->-+   .
     U_k     ^   v     U_k                       ^   v
         .   +   +   .                       .   +   +   .
             |   |                               |   |
For any k >= 0, U_k is a closed curve with length 4^(k+1) and visiting every lattice point (x, y) with 0 <= x, y < 2^(k+1).
The space filling curve U corresponds to the limit of U_k as k tends to infinity, and is a variant of H-order curve.
U visits once every lattice points with nonnegative coordinates and has a single connected component.

			Square array starts:
  n\k|    0    1    2    3    4    5    6    7
  ---+----------------------------------------
    0|    0....1....2....3    8....9...10...11
     |    |              |    |              |
    1|   -1   -6...-7    4    7   18...17   12
     |    |    |    |    |    |    |    |    |
    2|   -2   -5   -8    5....6   19   16   13
     |    |    |    |              |    |    |
    3|   -3...-4   -9  -26..-27   20   15...14
     |              |    |    |    |
    4|  -12..-11..-10  -25  -28   21...22...23
     |    |              |    |              |
    5|  -13  -18..-19  -24  -29  -34..-35   24
     |    |    |    |    |    |    |    |    |
    6|  -14  -17  -20  -23  -30  -33  -36   25..
     |    |    |    |    |    |    |    |
    7|  -15..-16  -21..-22  -31..-32  -37 -102..
     |                                  |    |

Rémy Sigrist, Table of n, a(n) for n = 0..5049
Rémy Sigrist, Representation of U_k for k = 0..5
Rémy Sigrist, Colored representation of U_7
Rémy Sigrist, Colored representation of the table for 0 <= x, y, <= 1023 (where the hue is function of T(y, x))
Rémy Sigrist, PARI program for A334188

See A163334, A323335 and A334232 for similar sequences.

See A334220, A334221, A334222 and A334223 for the coordinates of the curve.

PARI
```
See Links section.
```

A163897 a(n) = A163531(n)-A163547(n).

Original entry on oeis.org

0, 0, 2, 4, -2, -8, -6, 0, 0, 0, 2, 16, 16, 12, 6, 0, 0, 8, 10, 24, 28, 16, 0, 0, 0, 0, 10, 28, 24, 16, 32, 40, 48, 48, 24, 20, -2, -24, -40, -36, -40, -44, -64, -60, -56, -48, -42, -56, -42, -36, -20, -16, -8, 0, 16, 8, 14, 36, 34, 24, 28, 28, 18, 24, 16, 8, -8, 0, -10

Offset: 0

Antti Karttunen, Sep 19 2009

sign

This sequence gives the difference of squares of distance from the origin to the n-th term, in the Peano curve (A163334) and the Hilbert curve (A163357) on an N x N grid. Because the Hilbert curve is based on powers of 4 and the Peano curve on powers of 9, the graph of this sequence contains dramatic swings. [Edited to Peano vs Hilbert by Kevin Ryde, Aug 28 2020]

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

Cf. A165480 (positions of 0's).

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))

cp's OEIS Frontend

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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A163359 Hilbert curve in N x N grid, starting downwards 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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A147995 Array of N X N grid hopping "almost-walk", read by antidiagonals.

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A163358 Inverse permutation to A163357.

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A163328 Square array A, where entry A(y,x) has the ternary digits of x interleaved with the ternary digits of y, 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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A057212 n-th run has length n.

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A334188 T(n, k) is the number of steps from the point (0, 0) to the point (k, n) along the space filling curve U described in Comments section; a negative value corresponds to moving backwards; square array T(n, k), n, k >= 0 read by antidiagonals downwards.

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