A213197 T(n,k) = (2*(n+k)^2 - 2*(n+k) - 4*k + 6 + (2*k-2)*(-1)^n + (2*k-1)*(-1)^k + (-2*n+1)*(-1)^(n+k))/4; n, k > 0, read by antidiagonals.

1, 3, 4, 2, 6, 5, 8, 9, 11, 12, 7, 15, 10, 14, 13, 17, 18, 20, 21, 23, 24, 16, 28, 19, 27, 22, 26, 25, 30, 31, 33, 34, 36, 37, 39, 40, 29, 45, 32, 44, 35, 43, 38, 42, 41, 47, 48, 50, 51, 53, 54, 56, 57, 59, 60, 46, 66, 49, 65, 52, 64, 55, 63, 58, 62, 61, 68

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). Let m be natural number. The order of the list:

T(1,1)=1;

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

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

...

T(1,2*m+1), T(1,2*m), T(2, 2*m-1), T(3, 2*m-1),... T(2*m,1), T(2*m+1,1);

T(2*m,2), T(2*m-2,4), ...T(2,2*m);

...

Movement along two adjacent antidiagonals. The first row consists of phases: step to the west, step to the southwest, step to the south. The second row consists of phases: 2 steps to the north, 2 steps to the east. The length of each step is 1.

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

Formula

As a table:

T(n,k) = (2*(n+k)^2 - 2*(n+k) - 4*k + 6 + (2*k-2)*(-1)^n + (2*k-1)*(-1)^k + (-2*n+1)*(-1)^(n+k))/4.

As a linear sequence:

a(n) = (2*A003057(n)^2 - 2*A003057(n) - 4*A004736(n) + 6 + (2*A004736(n)-2)*(-1)^A002260(n) + (2*A004736(n)-1)*(-1)^A004736(n) + (-2*A002260(n)+1)*(-1)^A003056(n))/4;

a(n) = (2*(t+2)^2 - 2*(t+2) - 4*j + 6 + (2*j-2)*(-1)^i + (2*j-1)*(-1)^j + (-2*i+1)*(-1)^t)/4, where i = n - t*(t+1)/2, j = (t*t + 3*t + 4)/2 - n, t = floor((-1+sqrt(8*n-7))/2).

Example

The start of the sequence as a table:
   1,  3,  2,  8,  7, 17, 16, ...
   4,  6,  9, 15, 18, 28, 31, ...
   5, 11, 10, 20, 19, 33, 32, ...
  12, 14, 21, 27, 34, 44, 51, ...
  13, 23, 22, 36, 35, 53, 52, ...
  24, 26, 37, 43, 54, 64, 75, ...
  25, 39, 38, 56, 55, 77, 76, ...
  ...
The start of the sequence as a triangular array read by rows:
   1;
   3,  4;
   2,  6,  5;
   8,  9, 11, 12;
   7, 15, 10, 14, 13;
  17, 18, 20, 21, 23, 24;
  16, 28, 19, 27, 22, 26, 25;
  ...
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 2*r-2 of the triangular array above.
Last  2*r-1 numbers are from row 2*r-1 of the triangular array above.
   1;
   3,  4,  2,  6,  5;
   8,  9, 11, 12,  7, 15, 10, 14, 13;
  17, 18, 20, 21, 23, 24, 16, 28, 19, 27, 22, 26, 25;
  ...
Row r contains permutation of 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-2*r+2, 2*r*r-2*r+1.

Maple

T:=(n, k)->(2*(n+k)^2-2*(n+k)-4*k+6+(2*k-2)*(-1)^n+(2*k-1)*(-1)^k+(1-+2*n)*(-1)^(n+k))/4: seq(seq(T(k, n-k), k=1..n-1), n=1..13); # Muniru A Asiru, Dec 06 2018

Mathematica

T[n_, k_] := (2(n+k)^2 - 2(n+k) - 4k + 6 + (2k-2)(-1)^n + (2k-1)(-1)^k + (-2n+1)(-1)^(n+k))/4;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 06 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=(2*(t+2)**2-2*(t+2)-4*j+6 +(2*j-2)*(-1)**i+(2*j-1)*(-1)**j+(-2*i+1)*(-1)**t)/4

Crossrefs

Cf. A002260, A004736, A003056, A003057.

Table T(n,k) contains: in rows A130883, A033816, A100037, A000384, A100038, A014106, A091823; in columns A001844, A142463, A090288, A139570, A046092, A051890, A059993, A097080, A181510, A137882, A152813.

Sequence in context: A247413 A108127 A207376 * A378127 A049277 A214917

Adjacent sequences: A213194 A213195 A213196 * A213198 A213199 A213200

Keyword

nonn,tabl

Author

Boris Putievskiy, Mar 01 2013

Status

approved