A213927 T(n,k) = (z*(z-1)-(-1+(-1)^(z^2 mod 3))*n+(1+(-1)^(z^2 mod 3))*k)/2, where z=n+k-1; n, k > 0, read by antidiagonals.

1, 2, 3, 6, 5, 4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 21, 20, 19, 18, 17, 16, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 45, 44, 43, 42, 41, 40, 39, 38, 37, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 78

Offset: 1

Author

Boris Putievskiy, Mar 06 2013

Keywords

• nonn
• tabl
• uned

Comments

Self-inverse 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.

In general, let b(z) be a sequence of integers and denote number of antidiagonal table T(n,k) by z=n+k-1. Natural numbers placed in table T(n,k) by antidiagonals. The order of placement - by antidiagonal downwards, if b(z) is odd; by antidiagonal upwards, if b(z) is even. T(n,k) read by antidiagonals downwards. For A218890 -- the order of placement -- at the beginning m antidiagonals downwards, next m antidiagonals upwards and so on - b(z)=floor((z+m-1)/m). For this sequence b(z)=z^2 mod 3. (This comment should be edited for clarity, Joerg Arndt, Dec 11 2014)

Examples

			The start of the sequence as table.
The direction of the placement denoted by ">" and  "v".
.v.....v       v...v        v....v
.1.....2...6...7..11...21...22...29...45...
.3.....5...8..12..20...23...30...44...47...
>4.....9..13..19..24...31...43...48...58...
.10...14..18..25..32...42...49...59...75...
.15...17..26..33..41...50...60...74...83...
>16...27..34..40..51...61...73...84...97...
.28...35..39..52..62...72...85...98..114...
.36...38..53..63..71...86...99..113..128...
>37...54..64..70..87..100..112..129..145...
...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   6,  5,  4;
   7,  8,  9, 10;
  11, 12, 13, 14, 15;
  21, 20, 19, 18, 17, 16;
  22, 23, 24, 25, 26, 27, 28;
  29, 30, 31, 32, 33, 34, 35, 36;
  45, 44, 43, 42, 41, 40, 39, 38, 37;
  ...
Row r consists of r consecutive numbers from r*r/2-r/2+1 to r*r/2+r.
If r is not divisible by 3, rows are increasing.
If r is     divisible by 3, rows are decreasing.
		

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

Crossrefs

Cf. A218890, A056011, A056023, A130196, A011655, A001651, A008585.

Programs

Mathematica

T[n_, k_] := With[{z = n + k - 1}, (z*(z - 1) - (-1 + (-1)^Mod[z^2, 3])*n + (1 + (-1)^Mod[z^2, 3])*k)/2];
Table[T[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jul 22 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
z=i+j-1
result=(z*(z-1)-(-1+(-1)**(z**2%3))*i+(1+(-1)**(z**2%3))*j)/2

Formula

For the general case.

T(n,k) = (z*(z-1)-(-1+(-1)^b(z))*n+(1+(-1)^b(z))*k)/2, where z=n+k-1 (as a table).

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

For this sequence b(z)=z^2 mod 3.

T(n,k) = (z*(z-1)-(-1+(-1)^(z^2 mod 3))*n+(1+(-1)^(z^2 mod 3))*k)/2, where z=n+k-1 (as a table).

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