OFFSET
1,2
COMMENTS
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(1,3), T(1,2), T(2,1), T(2,2), T(3,1);
...
T(1,n), T(1,n-1), T(2,n-2), T(2,n-1), T(3,n-2), T(3,n-3)...T(n,1);
...
Descent by snake along two adjacent antidiagonal - step to the west, step to the southwest, step to the east, step to the southwest and so on. The length of each step is 1.
Table contains:
row 2 accommodates elements A100037 in odd places;
column 3 accommodates elements A130861 in even places;
main diagonal accommodates elements A188135 in odd places,
LINKS
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
FORMULA
As a table:
T(n,k) = ((k + n)^2 - 4*k + 3 + (-1)^k - (k + n - 2)*(-1)^(k + n))/2.
As a linear sequence:
a(n) = ((t + 2)^2 - 4*j + 3 + (-1)^j - t*(-1)^t)/2, where j = (t*t + 3*t + 4)/2 - n and t = int((sqrt(8*n - 7) - 1)/ 2).
EXAMPLE
The start of the sequence as a table:
1, 3, 2, 8, 7, 17, 16, 30, 29, 47, 46, ...
4, 5, 9, 10, 18, 19, 31, 32, 48, 49, 69, ...
6, 12, 11, 21, 20, 34, 33, 51, 50, 72, 71, ...
13, 14, 22, 23, 35, 36, 52, 53, 73, 74, 98, ...
15, 25, 24, 38, 37, 55, 54, 76, 75, 101, 100, ...
26, 27, 39, 40, 56, 57, 77, 78, 102, 103, 131, ...
28, 42, 41, 59, 58, 80, 79, 105, 104, 134, 133, ...
43, 44, 60, 61, 81, 82, 106, 107, 135, 136, 168, ...
45, 63, 62, 84, 83, 109, 108, 138, 137, 171, 170, ...
64, 65, 85, 86, 110, 111, 139, 140, 172, 173, 209, ...
66, 88, 87, 113, 112, 142, 141, 175, 174, 212, 211, ...
...
The start of the sequence as triangle array read by rows:
1;
3, 4;
2, 5, 6;
8, 9, 12, 13;
7, 10, 11, 14, 15;
17, 18, 21, 22, 25, 26;
16, 19, 20, 23, 24, 27, 28;
30, 31, 34, 35, 38, 39, 42, 43;
29, 32, 33, 36, 37, 40, 41, 44, 45;
47, 48, 51, 52, 55, 56, 59, 60, 63, 64;
46, 49, 50, 53, 54, 57, 58, 61, 62, 65, 66;
...
The start of the sequence as an array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from row number 2*r-2 of the triangular array above.
Last 2*r-1 numbers are from row number 2*r-1 of the triangular array above.
1;
3, 4, 2, 5, 6;
8, 9, 12, 13, 7, 10, 11, 14, 15;
17, 18, 21, 22, 25, 26, 16, 19, 20, 23, 24, 27, 28;
30, 31, 34, 35, 38, 39, 42, 43, 29, 32, 33, 36, 37, 40, 41, 44, 45;
47, 48, 51, 52, 55, 56, 59, 60, 63, 64, 46, 49, 50, 53, 54, 57, 58, 61, 62, 65, 66;
...
Row number r contains permutation 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-5*r+5, 2*r*r-5*r+6,...2*r*r-r-4, 2*r*r-r-1, 2*r*r-r.
MATHEMATICA
T[n_, k_] := ((k+n)^2 - 4k + 3 + (-1)^k - (k+n-2)(-1)^(k+n))/2;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 29 2018 *)
PROG
(Python)
t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=((t+2)**2-4*j+3+(-1)**j-t*(-1)**(t+2))/2
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Boris Putievskiy, Feb 07 2013
STATUS
approved