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COMMENTS
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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.
Enumeration table T(n,k). The order of the list:
T(1,1)=1;
T(2,1), T(2,2), T(1,2), T(1,3), T(3,1),
...
T(2,n-1), T(4,n-3), T(6,n-5), ..., T(n,1),
T(2,n), T(4,n-2), T(6,n-4), ..., T(n,2),
T(1,n), T(3,n-2), T(5,n-4), ..., T(n-1,2),
T(1,n+1), T(3,n-1), T(5,n-3), ..., T(n+1,1),
...
The order of the list elements of adjacent antidiagonals. Let m be a positive integer.
Movement by antidiagonal {T(1,2*m), T(2*m,1)} from T(2,2*m-1) to T(2*m,1) length of step is 2,
movement by antidiagonal {T(1,2*m+1), T(2*m+1,1)} from T(2,2*m) to T(2*m,2) length of step is 2,
movement by antidiagonal {T(1,2*m), T(2*m,1)} from T(1,2*m) to T(2*m-1,2) length of step is 2,
movement by antidiagonal {T(1,2*m+1), T(2*m+1,1)} from T(1,2*m+1) to T(2*m+1,1) length of step is 2.
Table contains:
row 5 accommodates elements A097080 in odd places,
row 7 accommodates elements A137882 in odd places,
row 10 accommodates elements A100038 in odd places,
row 14 accommodates elements A100039 in odd places;
column 7 accommodates elements A071355 in odd places,
column 9 accommodates elements |A147973| in even places,
column 10 accommodates elements A139570 in odd places,
column 13 accommodates elements A130861 in odd places.
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EXAMPLE
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The start of the sequence as table:
1, 4, 5, 11, 13, 22, 25, 37, 41, 56, 61, ...
2, 3, 7, 9, 16, 19, 29, 33, 46, 51, 67, ...
6, 12, 14, 23, 26, 38, 42, 57, 62, 80, 86, ...
8, 10, 17, 20, 30, 34, 47, 52, 68, 74, 93, ...
15, 24, 27, 39, 43, 58, 63, 81, 87, 108, 115, ...
18, 21, 31, 35, 48, 53, 69, 75, 94, 101. 123, ...
28, 40, 44, 59, 64, 82, 88, 109, 116, 140, 148, ...
32, 36, 49, 54, 70, 76, 95, 102, 124, 132, 157, ...
45, 60, 65, 83, 89, 110, 117, 141, 149, 176, 185, ...
50, 55, 71, 77, 96, 103, 125, 133, 158, 167, 195, ...
66, 84, 90, 111, 118, 142, 150, 177, 186, 216, 226, ...
...
The start of the sequence as triangle array read by rows:
1;
4, 2;
5, 3, 6;
11, 7, 12, 8;
13, 9, 14, 10, 15;
22, 16, 23, 17, 24, 18;
25, 19, 26, 20, 27, 21, 28;
37, 29, 38, 30, 39, 31, 40, 32;
41, 33, 42, 34, 43, 35, 44, 36, 45;
56, 46, 57, 47, 58, 48, 59, 49, 60, 50;
61, 51, 62, 52, 63, 53, 64, 54, 65, 55, 66;
...
The start of the sequence as array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from the row number 2*r-2 of triangle array, located above.
Last 2*r-1 numbers are from the row number 2*r-1 of triangle array, located above.
1;
4, 2, 5, 3, 6;
11, 7,12, 8,13, 9,14,10,15;
22,16,23,17,24,18,25,19,26,20,27,21,28;
37,29,38,30,39,31,40,32,41,33,42,34,43,35,44,36,45;
56,46,57,47,58,48,59,49,60,50,61,51,62,52,63,53,64,54,65,55,66;
...
Row number r contains permutation numbers 4*r-3 from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-3*r+2,2*r*r-5*r+4, 2*r*r-3*r+3, 2*r*r-5*r+5, 2*r*r-3*r+4, 2*r*r-5*r+6, ..., 2*r*r-3*r+1, 2*r*r-r.
...
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