A210521 Array read by downward antidiagonals: T(n,k) = (n+k-1)*(n+k-2) + n + floor((n+k)/2)*(1-2*floor((n+k)/2)), for n, k > 0

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

Offset

1,2

Comments

Enumeration table T(n,k). The order of the list: T(1,1)=1; for k>0: T(1,2*k+1),T(1,2*k); T(2,2*k),T(2,2*k-1); ... T(2*k,2),T(2*k,1); T(2*k+1,1).

The order of the list is descent stairs from the northeast to southwest: step to the west, step to the south, step to the west and so on. The length of each step is 1 or alternation of elements pair adjacent antidiagonals.

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.

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

Index entries for sequences that are permutations of the natural numbers

Formula

As a table: T(n,k) = (n+k-1)*(n+k-2) + 2*n + floor((n+k)/2)*(1-2*floor((n+k)/2)).

As a linear sequence: a(n) = 2*A000027(n) + A204164(n)*(1-2*A204164(n)).

a(n) = 2*n+v*(1-2*v), where t = floor((-1+sqrt(8*n-7))/2) and v = floor((t+2)/2).

G.f. as a table: (2 - 2*y - 5*y^2 + 6*y^3 + 3*y^4 + x*y*(1 + 3*y-5*y^2 + y^3) + x^2*(- 3 + 7*y + 5*y^2 - 11*y^3 - 6*y^4) - x^3*(- 4 + 5*y + 7*y^2 - 9*y^3 + y^4) + x^4*(1 - y - 4*y^2 + y^3 + 7*y^4))/((- 1 + x)^3*(1 + x)^2*(- 1 + y)^3*(1 + y)^2). - Stefano Spezia, Dec 03 2018

Example

The start of the sequence as a table:
   1,  3,  2,  8,  7,  17,  16,  30,  29, ...
   5,  4, 10,  9, 19,  18,  32,  31,  49, ...
   6, 12, 11, 21, 20,  34,  33,  51,  50, ...
  14, 13, 23, 22, 36,  35,  53,  52,  74, ...
  15, 25, 24, 38, 37,  55,  54,  76,  75, ...
  27, 26, 40, 39, 57,  56,  78,  77, 103, ...
  28, 42, 41, 59, 58,  80,  79, 105, 104, ...
  44, 43, 61, 60, 82,  81, 107, 106, 136, ...
  45, 63, 62, 84, 83, 109, 108, 138, 137, ...
  ...
The start of the sequence as a triangular array read by rows:
   1;
   3,  5;
   2,  4,  6;
   8, 10, 12, 14;
   7,  9, 11, 13, 15;
  17, 19, 21, 23, 25, 27;
  16, 18, 20, 22, 24, 26, 28;
  30, 32, 34, 36, 38, 40, 42, 44;
  29, 31, 33, 35, 37, 39, 41, 43, 45;
  ...
The sequence as array read by rows, the length of row r is 4*r-1. First 2*r-1 numbers are from row 2*r-1 of the triangular array above. Last 2*r numbers are from row 2*r of the triangular array. The start of the sequence:
1,3,5;
2,4,6,8,10,12,14;
7,9,11,13,15,17,19,21,23,25,27;
16,18,20,22,24,26,28,30,32,34,36,38,40,42,44;
29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65;
...
Row r contains 4*r-1 numbers: 2*r^2-5*r+4, 2*r^2-5*r+6, 2*r^2-5*r+8, ..., r*(2*r+3).
Considered as a triangle, the rows have constant parity.

Mathematica

T[n_, k_] := (n+k-1)(n+k-2) + 2n + Floor[(n+k)/2](1 - 2 Floor[(n+k)/2]);
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 03 2018 *)

Prog

(Python)
t=int((math.sqrt(8*n-7)-1)/2)
v=int((t+2)/2)
result=2*n+v*(1-2*v)

Crossrefs

Cf. A000027, A204164, the main diagonal is A084849.

Sequence in context: A309492 A131793 A065186 * A219249 A203553 A081964

Adjacent sequences: A210518 A210519 A210520 * A210522 A210523 A210524

Keyword

nonn,tabl

Author

Boris Putievskiy, Jan 26 2013

Status

approved