A060734 -id:A060734 - cp's OEIS frontend

A064790 Inverse permutation to A060734.

1, 3, 5, 2, 6, 9, 13, 8, 4, 10, 14, 19, 25, 18, 12, 7, 15, 20, 26, 33, 41, 32, 24, 17, 11, 21, 27, 34, 42, 51, 61, 50, 40, 31, 23, 16, 28, 35, 43, 52, 62, 73, 85, 72, 60, 49, 39, 30, 22, 36, 44, 53, 63, 74, 86, 99, 113, 98, 84, 71, 59, 48, 38, 29, 45, 54, 64, 75, 87, 100, 114

Offset: 1

N. J. A. Sloane, Oct 20 2001

From Boris Putievskiy, Mar 14 2013: (Start)
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and  from T(n,n) to T(n,1). This sequence is A188568 as table read  by boustrophedonic ("ox-plowing") method - layer clockwise, layer counterclockwise and so. The same table A188568 read layer by layer clockwise is A194280. (End)

			From _Boris Putievskiy_, Mar 14 2013: (Start)
The start of the sequence as table:
  1....2...6...7..15..16..28...
  3....5...9..12..20..23..35...
  4....8..13..18..26..31..43...
  10..14..19..25..33..40..52...
  11..17..24..32..41..50..62...
  21..27..34..42..51..61..73...
  22..30..39..49..60..72..85...
  ...
The start of the sequence as triangular array read by rows:
  1;
  3,5,2;
  6,9,13,8,4;
  10,14,19,25,18,12,7;
  15,20,26,33,41,32,24,17,11;
  21,27,34,42,51,61,50,40,31,23,16;
  28,35,43,52,62,73,85,72,60,49,39,30,22;
  ...
Row number r contains 2*r-1 numbers. (End)

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's MathWorld, Pairing Function
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A064788, A188568, A194280.

a(n) = (i+j-1)*(i+j-2)/2+i, where i=min(t; t^2-n+1), j=min(t; n-(t-1)^2), t=floor(sqrt(n-1))+1. - Boris Putievskiy, Dec 24 2012

A194195 First inverse function (numbers of rows) for pairing function A060734.

Original entry on oeis.org

1, 2, 2, 1, 3, 3, 3, 2, 1, 4, 4, 4, 4, 3, 2, 1, 5, 5, 5, 5, 5, 4, 3, 2, 1, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1, 7, 7, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 1, 8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 5, 4, 3, 2, 1, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 1

Boris Putievskiy, Dec 21 2012

The sequence is the second inverse function (numbers of columns) for pairing function A060736.

			The start of the sequence as triangle array read by rows:
1;
2,2,1;
3,3,3,2,1;
4,4,4,4,3,2,1;
. . .
Row number k contains 2k-1 numbers k,k,...k,k-1,k-2,...1 (k times repetition "k").

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A060734, A060736, A220603, A220604

f[n_]:=Module[{t=Floor[Sqrt[n-1]]+1},Min[t,t^2-n+1]]; Array[f,80] (* Harvey P. Dale, Dec 31 2012 *)

t=int(math.sqrt(n-1)) +1
i=min(t,t**2-n+1)

a(n) = min{t; t^2 - n + 1}, where t=floor(sqrt(n-1))+1.

A194258 Second inverse function (numbers of columns) for pairing function A060734.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 3, 3, 3, 1, 2, 3, 4, 4, 4, 4, 1, 2, 3, 4, 5, 5, 5, 5, 5, 1, 2, 3, 4, 5, 6, 6, 6, 6, 6, 6, 1, 2, 3, 4, 5, 6, 7, 7, 7, 7, 7, 7, 7, 1, 2, 3, 4, 5, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 1, 2, 3, 4, 5, 6

Offset: 1

Boris Putievskiy, Dec 21 2012

The sequence is the first inverse function (numbers of rows) for pairing function A060736.

			The start of the sequence as triangle array read by rows:
1;
1,2,2;
1,2,3,3,3;
1,2,3,4,4,4,4;
. . .
Row number k contains 2k-1 numbers 1,2,...k-1,k,k,...k (k times repetition "k").

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A060734, A060736, A220603, A220604.

Flatten[Table[Join[Range[n-1],Table[n,{n}]],{n,10}]] (* Harvey P. Dale, Jun 23 2013 *)

t=int(math.sqrt(n-1)) +1
j=min(t,n-(t-1)**2)

a(n) = min{t; n - (t - 1)^2}, where t=floor(sqrt(n-1))+1.

A060736 Array of square numbers read by antidiagonals in up direction.

Original entry on oeis.org

1, 2, 4, 5, 3, 9, 10, 6, 8, 16, 17, 11, 7, 15, 25, 26, 18, 12, 14, 24, 36, 37, 27, 19, 13, 23, 35, 49, 50, 38, 28, 20, 22, 34, 48, 64, 65, 51, 39, 29, 21, 33, 47, 63, 81, 82, 66, 52, 40, 30, 32, 46, 62, 80, 100

Offset: 1

Frank Ellermann, Apr 23 2001

A simple permutation of natural numbers.

a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers. - Boris Putievskiy, Jan 09 2013

			1 4 9 16 .. => a(1)= 1
2 3 8 15 .. => a(2)= 2, a(3)=4
5 6 7 14 .. => a(4)= 5, a(5)=3, a(6)=9
10 11 12 13 .. => a(7)=10, a(8)=6, a(9)=8, a(10)=16

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734. Inverse permutation: A064788, the first inverse function (numbers of rows) A194258, the second inverse function (numbers of columns) A194195.

Table[ If[n < 2*k-1, k^2 + k - n, (n-k)^2 + k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 09 2013 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i>=j:
   result=i**2-j+1
else:
   result=(j-1)**2+i
# Boris Putievskiy, Jan 09 2013

T(n+1, k)=n*n+k, T(k, n+1)=(n+1)*(n+1)+1-k, 1 <= k <= n+1.

a(n)=i^2-j+1 if i >= j, a(n)=(j-1)^2 + i if i < j, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Jan 09 2013

A326407 Minesweeper sequence of positive integers arranged on a 2D grid along a square array that grows by alternately adding a row at its bottom edge and a column at its right edge.

Original entry on oeis.org

2, -1, -1, 2, -1, 5, -1, 2, 1, 3, -1, 4, -1, 3, 2, 0, -1, 3, -1, 3, 3, 2, -1, 1, 0, 2, 3, 2, -1, 3, -1, 1, 2, 2, 2, 0, -1, 1, 2, 3, -1, 3, -1, 3, 3, 2, -1, 1, 0, 1, 2, 2, -1, 2, 3, 2, 3, 2, -1, 2, -1, 3, 2, 0, 1, 2, -1, 2, 1, 1, -1, 3, -1, 1, 1, 1, 3, 3, -1, 1, 0

Offset: 1

Witold Tatkiewicz, Oct 02 2019

Place positive integers on a 2D grid starting with 1 in the top left corner and continue along an increasing square array as in A060734.
Replace each prime with -1 and each nonprime with the number of primes in adjacent grid cells around them.
n is replaced by a(n).
This sequence treats prime numbers as "mines" and fills gaps according to the rules of the classic Minesweeper game.
a(n) < 6 (conjectured).
a(n) = 5 for n={6} (conjectured).
Set of n such that a(n) = 4 is unbounded (conjectured).

			Consider positive integers distributed onto the plane along an increasing square array:
   1  4  9 16 25 36
   2  3  8 15 24 35
   5  6  7 14 23 34
  10 11 12 13 22 33
  17 18 19 20 21 32
  26 27 28 29 30 31
...
1 is not prime and in adjacent grid cells there are 2 primes: 2 and 3. Therefore a(1) = 2.
2 is prime, therefore a(2) = -1.
8 is not prime and in adjacent grid cells there are 2 primes: 3, and 7. Therefore a(8) = 2.
Replacing n with a(n) in the plane described above, and using "." for a(n) = 0 and "*" for negative a(n), we produce a graph resembling Minesweeper, where the mines are situated at prime n:
  2  2  1  .  .  .  .  .  .  .  .  . ...
  *  *  2  2  1  2  1  2  1  1  .  1
  *  5  *  3  *  2  *  3  *  2  1  1
  3  *  4  *  2  2  2  *  3  *  1  1
  *  3  *  3  3  1  3  2  3  1  2  1
  2  3  2  *  3  *  3  *  1  .  1  *
  *  1  2  3  *  3  *  2  1  .  2  3
  1  2  2  *  2  3  2  3  1  2  2  *
  1  2  *  2  1  1  *  3  *  2  *  2
  2  *  3  2  .  2  3  *  3  3  1  1
  *  3  *  1  1  2  *  3  *  2  1  .
  1  2  1  2  2  *  3  3  2  *  1  1
...
In order to produce the sequence, the graph is read along the original mapping.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Minesweeper-style graph read along original mapping, replacing -1 with a "mine", and 0 with blank space.
Michael De Vlieger, Square plot of a million terms read along original mapping, with black indicating a prime and levels of gray commensurate to a(n).
Witold Tatkiewicz, link for Java program
Wikipedia, Minesweeper game

Cf. A060734 - plane mapping
Different arrangements of integers:
Cf. A326405 - antidiagonals,
Cf. A326406 - triangle maze,
Cf. A326408 - square maze,
Cf. A326409 - Hamiltonian path,
Cf. A326410 - Ulam's spiral.

Java
```
See Links section.
```

Block[{n = 12, s}, s = ArrayPad[Array[If[#1 < 2 #2 - 1, #2^2 + #2 - #1, (#1 - #2)^2 + #2] & @@ {#1 + #2 - 1, #2} &, {# + 1, # + 1}], 1] &@ n; Table[If[PrimeQ@ m, -1, Count[#, ?PrimeQ] &@ Union@ Map[s[[#1, #2]] & @@ # &, Join @@ Array[FirstPosition[s, m] + {##} - 2 &, {3, 3}]]], {m, PolygonalNumber@ n}]] (* _Michael De Vlieger, Oct 02 2019 *)

A185725 Array associated with squares, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 10, 7, 8, 11, 17, 12, 9, 13, 18, 26, 19, 14, 15, 20, 27, 37, 28, 21, 16, 22, 29, 38, 50, 39, 30, 23, 24, 31, 40, 51, 65, 52, 41, 32, 25, 33, 42, 53, 66, 82, 67, 54, 43, 34, 35, 44, 55, 68, 83, 101, 84, 69, 56, 45, 36, 46, 57, 70, 85, 102, 122, 103, 86, 71, 58, 47, 48, 59, 72, 87, 104, 123, 145, 124, 105, 88, 73, 60, 49, 61, 74, 89, 106, 125, 146, 170, 147, 126, 107, 90, 75, 62, 63, 76, 91, 108, 127, 148, 171

Offset: 1

Clark Kimberling, Feb 01 2011

Every positive integer occurs exactly once; hence, as a sequence, A185725 is a permutation of the positive integers. The square with corners T(0,0)=1 and T(n,n)=n^2 is occupied by the numbers 1,2,...,n^2.

			Northwest corner:
1...2...5...10...17
3...4...7...12...19
6...8...9...14...21
11..13..15..16...23

Cf. A185726, A060734, A185728.

f[n_,k_]:=(k-1)^2+2*n-1/; n<=k;
f[n_,k_]:=(n-1)^2+2*k/; n>k;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k)=(k-1)^2+2n-1 if n<=k; T(n,k)=(n-1)^2+2k if n>k.

A185728 Array associated with squares, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 10, 6, 7, 15, 17, 11, 9, 14, 24, 26, 18, 12, 13, 23, 35, 37, 27, 19, 16, 22, 34, 48, 50, 38, 28, 20, 21, 33, 47, 63, 65, 51, 39, 29, 25, 32, 46, 62, 80, 82, 66, 52, 40, 30, 31, 45, 61, 79, 99, 101, 83, 67, 53, 41, 36, 44, 60, 78, 98, 120, 122, 102, 84, 68, 54, 42, 43, 59, 77, 97, 119, 143, 145, 123, 103, 85, 69, 55, 49, 58, 76, 96, 118, 142, 168, 170, 146, 124, 104, 86, 70, 56, 57, 75, 95, 117, 141, 167, 195

Offset: 1

Clark Kimberling, Feb 01 2011

Every positive integer occurs exactly once; hence, as a sequence, A185725 is a permutation of the positive integers.  The square with corners T(0,0)=1 and T(n,n)=n^2 is occupied by the numbers 1,2,...,n^2.
T(1,k)=(k-1)^2+1 (A002522)
T(n,1)=-1+n^2 for n>=2.

			Northwest corner:
1...2...5...10...17
3...4...6...11...18
8...7...9...12...19
15..14..13..16...20

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A185725, A185726, A060734.

f[n_,k_]:=(k-1)^2+n/; k>n;
f[n_,n_]:=n^2; f[n_,k_]:=n^2-k/; k

T(n,k)=(k-1)^2+n if nk.

A185726 Array associated with squares, by antidiagonals.

Original entry on oeis.org

1, 3, 4, 8, 10, 10, 18, 22, 24, 21, 35, 44, 45, 48, 39, 61, 80, 81, 84, 86, 66, 98, 134, 138, 136, 144, 142, 104, 148, 210, 222, 216, 220, 231, 220, 155, 213, 312, 339, 332, 325, 340, 351, 324, 221, 295, 444, 495, 492, 475, 480, 504, 510, 458, 304, 396, 610, 696, 704, 680, 666, 690, 720, 714, 626, 406, 518, 814, 948, 976, 950, 918, 924, 965, 996, 969, 832, 529, 663, 1060, 1257, 1316, 1295, 1248, 1225, 1260, 1315, 1340, 1281, 1080, 675, 833, 1352, 1629, 1732, 1725

Offset: 1

Clark Kimberling, Feb 01 2011

Every positive integer occurs exactly once; hence, as a sequence, A185725 is a permutation of the positive integers.  The square with corners T(0,0)=1 and T(n,n)=n^2 is occupied by the numbers 1,2,...,n^2.
T(n,1)=n^2 (A000290)
T(n,n)=(n-1)^2+1 (A002522)
T(1,k)=k^2-1 (A132411).

			Northwest corner:
1...3...8...15...24
4...2...6...13...22
9...7...5...11...20
16..14..12..10...18

Cf. A185725, A060734, A185728.

f[n_,k_]:=n^2-2*k+2/; n>=k;
f[n_,k_]:=k^2-2*n+1/; n

T(n,k)=n^2-2k+2 if n>=k; T(n,k)=k^2-2n+1 if n

A213921 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places clockwise. Table T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 8, 9, 13, 17, 14, 6, 16, 21, 26, 22, 11, 12, 25, 31, 37, 32, 18, 15, 20, 36, 43, 50, 44, 27, 23, 24, 30, 49, 57, 65, 58, 38, 33, 19, 35, 42, 64, 73, 82, 74, 51, 45, 28, 29, 48, 56, 81, 91, 101, 92, 66, 59, 39, 34, 41, 63, 72, 100, 111

Offset: 1

Boris Putievskiy, Mar 05 2013

A permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Enumeration table T(n,k) is layer by layer. The order of the list:
T(1,1)=1;
T(1,2), T(2,1), T(2,2);
. . .
T(1,n), T(3,n), ... T(n,3), T(n,1), T(2,n), T(4,n), ... T(n,4), T(n,2);
...

			The start of the sequence as table:
   1   2   5  10  17  26 ...
   3   4   8  14  22  32 ...
   7   9   6  11  18  27 ...
  13  16  12  15  23  33 ...
  21  25  20  24  19  28 ...
  31  36  30  35  29  34 ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   5,  4,  7;
  10,  8,  9, 13;
  17, 14,  6, 16, 21;
  26, 22, 11, 12, 25, 31;
  ...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736; table T(n,k) contains: in rows A002522, A014206, A059100, A027688, A117950, A027689, A087475, A027690, A117951, A027691, A114949, A027692, A117619; in columns A002061, A000290, A002378, A005563, A028387, A008865, A028552, A028872, A014209, A028347, A028875.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i > j:
   result=i*i-(j%2)*i+2-int((j+2)/2)
else:
   result=j*j-((i%2)+1)*j + int((i+3)/2)

As a table:
T(n,k) = n*n - (k mod 2)*n + 2 - floor((k+2)/2), if n>k;
T(n,k) = k*k - ((n mod 2)+1)*k + floor((n+3)/2), if n<=k.
As a linear sequence:
a(n) = i*i - (j mod 2)*i + 2 - floor((j+2)/2), if i>j;
a(n) = j*j - ((i mod 2)+1)*j + floor((i+3)/2), if i<=j; where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t = floor((-1+sqrt(8*n-7))/2).

A213922 Natural numbers placed in table T(n,k) layer by layer. The order of placement: T(n,n), T(n-1,n), T(n,n-1), ... T(1,n), T(n,1). Table T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 3, 4, 8, 2, 9, 15, 6, 7, 16, 24, 13, 5, 14, 25, 35, 22, 11, 12, 23, 36, 48, 33, 20, 10, 21, 34, 49, 63, 46, 31, 18, 19, 32, 47, 64, 80, 61, 44, 29, 17, 30, 45, 62, 81, 99, 78, 59, 42, 27, 28, 43, 60, 79, 100, 120, 97, 76, 57, 40, 26, 41, 58, 77, 98, 121

Offset: 1

Boris Putievskiy, Mar 05 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Enumeration table T(n,k) is layer by layer. The order of the list:
T(1,1)=1;
T(2,2), T(1,2), T(2,1);
...
T(n,n), T(n-1,n), T(n,n-1), ... T(1,n), T(n,1);
...

			The start of the sequence as a table:
   1,  3,  8, 15, 24, 35, ...
   4,  2,  6, 13, 22, 33, ...
   9,  7,  5, 11, 20, 31, ...
  16, 14, 12, 10, 18, 29, ...
  25, 23, 21, 19, 17, 27, ...
  36, 34, 32, 30, 28, 26, ...
...
The start of the sequence as triangular array read by rows:
   1;
   3,  4;
   8,  2,  9;
  15,  6,  7, 16;
  24, 13,  5, 14, 25;
  35, 22, 11, 12, 23, 36;
  ...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736; table T(n,k) contains: in rows A005563, A028872, A028875, A028881, A028560, A116711; in columns A000290, A008865, A028347, A028878, A028884.

f[n_, k_] := n^2 - 2*k + 2 /; n >= k; f[n_, k_] := k^2 - 2*n + 1 /; n < k; TableForm[Table[f[n, k], {n, 1, 5}, {k, 1, 10}]]; Table[f[n - k + 1, k], {n, 5}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Aug 19 2017 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i >= j:
   result=i*i-2*j+2
else:
   result=j*j-2*i+1

As a table,
  T(n,k) = n*n - 2*k + 2, if n >= k;
  T(n,k) = k*k - 2*n + 1, if n < k.
As a linear sequence,
  a(n) = i*i - 2*j + 2, if i >= j;
  a(n) = j*j - 2*i + 1, if i < j
  where
    i = n - t*(t+1)/2,
    j = (t*t + 3*t + 4)/2 - n,
    t = floor((-1 + sqrt(8*n-7))/2).

Showing 1-10 of 14 results. Next

cp's OEIS Frontend

A064790 Inverse permutation to A060734.

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A194195 First inverse function (numbers of rows) for pairing function A060734.

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A194258 Second inverse function (numbers of columns) for pairing function A060734.

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A060736 Array of square numbers read by antidiagonals in up direction.

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A326407 Minesweeper sequence of positive integers arranged on a 2D grid along a square array that grows by alternately adding a row at its bottom edge and a column at its right edge.

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A185725 Array associated with squares, by antidiagonals.

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A185728 Array associated with squares, by antidiagonals.

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A185726 Array associated with squares, by antidiagonals.