A213171 T(n,k) = ((k+n)^2 - 4*k + 3 - (-1)^n - (k+n)*(-1)^(k+n))/2; n, k > 0, read by antidiagonals.

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

Offset: 1

Author

Boris Putievskiy, Feb 14 2013

Keywords

• nonn
• tabl

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(2,2), T(1,2), T(2,1), T(3,1);

. . .

T(1,n), T(2,n-1), T(1,n-1), T(2,n-2), T(3,n-2), T(4,n-3)...T(n,1);

...

Descent by snake along two adjacent antidiagonals - step to the southwest, step to the north, step to the southwest, step to the south and so on. The length of each step is 1. Phase four steps is rotated 90 degrees counterclockwise and the mirror of the phase A211377.

Table contains the following:

row 1 is alternation of elements A130883 and A100037,

row 2 accommodates elements A033816 in even places;

column 1 is alternation of elements A000384 and A014106,

column 2 is alternation of elements A091823 and A071355,

column 4 accommodates elements A130861 in odd places;

main diagonal is alternation of elements A188135 and A033567,

diagonal 1, located above the main diagonal, accommodates elements A033585 in even places,

diagonal 2, located above the main diagonal, accommodates elements A139271 in odd places,

diagonal 3, located above the main diagonal, is alternation of elements A033566 and A194431.

Examples

			The start of the sequence as a table:
   1   4   2   9   7   8  16 ...
   5   3  10   8  19  17  32 ...
   6  13  11  22  20  35  33 ...
  14  12  23  21  36  34  53 ...
  15  26  24  39  37  56  54 ...
  27  25  40  38  57  55  78 ...
  28  43  41  60  58  81  79 ...
  ...
The start of the sequence as a triangle array read by rows:
   1
   4  5
   2  3  6
   9 10 13 14
   7  8 11 12 15
  18 19 22 23 26 27
  16 17 20 21 24 25 28
  ...
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  5  2  3  6
   9 10 13 14  7  8 11 12 15
  18 19 22 23 26 27 16 17 20 21 24 25 28
  ...
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+6, 2*r*r-5*r+7, ..., 2*r*r-r-4, 2*r*r-r-3, 2*r*r-r.
		

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 Weisstein's World of Mathematics, Pairing functions
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000384, A002260, A003056, A003057, A004736, A014106, A033566, A033567, A033585, A033816, A071355, A091823, A100037, A130861, A130883, A139271, A188135, A194431, A211377.

Programs

Maple

T:=(n,k)->((k+n)^2-4*k+3-(-1)^n-(k+n)*(-1)^(k+n))/2: seq(seq(T(k,n-k),k=1..n-1),n=1..13); # Muniru A Asiru, Dec 06 2018

Mathematica

T[n_, k_] := ((n+k)^2 - 4k + 3 - (-1)^n - (-1)^(n+k)(n+k))/2;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 06 2018 *)

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)**i-(t+2)*(-1)**t)/2

Formula

As a table:

T(n,k) = ((k+n)^2-4*k+3-(-1)^n-(k+n)*(-1)^(k+n))/2.

As a linear sequence:

a(n) = (A003057(n)^2-4*A004736(n)+3-(-1)^A002260(n)-A003057(n)*(-1)^A003056(n))/2;

a(n) = ((t+2)^2-4*j+3-(-1)^i-(t+2)*(-1)^t)/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).