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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified February 16 20:45 EST 2019. Contains 320189 sequences. (Running on oeis4.)