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

1,2

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)

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

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).

Example

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.

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 *)

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
z=i+j-1
result=(z*(z-1)-(-1+(-1)**(z**2%3))*i+(1+(-1)**(z**2%3))*j)/2

Crossrefs

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

Sequence in context: A277330 A072298 A130686 * A222241 A056023 A133259

Adjacent sequences: A213924 A213925 A213926 * A213928 A213929 A213930

Keyword

nonn,tabl,uned

Author

Boris Putievskiy, Mar 06 2013

Status

approved