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A188568 Enumeration table T(n,k) by diagonals. The order of the list - if n is odd:  T(n,1), T(2,n-1), T(n-2,3),..., T(n-1,2), T(1,n); if n is even: T(1,n), T(n-1,2), T(3,n-2),..., T(2,n-1), T(n,1). 7
1, 2, 3, 6, 5, 4, 7, 9, 8, 10, 15, 12, 13, 14, 11, 16, 20, 18, 19, 17, 21, 28, 23, 26, 25, 24, 27, 22, 29, 35, 31, 33, 32, 34, 30, 36, 45, 38, 43, 40, 41, 42, 39, 44, 37, 46, 54, 48, 52, 50, 51, 49, 53, 47, 55 (list; 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.

Call a "layer" a pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). This table read layer by layer clockwise is A194280. This table read by boustrophedonic ("ox-plowing") method - layer clockwise, layer counterclockwise and so on - is A064790. - Boris Putievskiy, Mar 14 2013

LINKS

Boris Putievskiy, Rows n = 1..140 of triangle, flattened

Boris Putievskiy, Transformations 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

a(n) = ((i+j-1)*(i+j-2)+((-1)^max(i,j)+1)*i-((-1)^max(i,j)-1)*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].

EXAMPLE

The start of the sequence as table:

   1,  2,  6,  7, 15, 16, 28, ...

   3,  5,  9, 12, 20, 23, 35, ...

   4,  8, 13, 18, 26, 31, 43, ...

  10, 14, 19, 25, 33, 40, 52, ...

  11, 17, 24, 32, 41, 50, 62, ...

  21, 27, 34, 42, 51, 61, 73, ...

  22, 30, 39, 49, 60, 72, 85, ...

  ...

The start of the sequence as triangular array read by rows:

   1;

   2,  3;

   6,  5,  4;

   7,  9,  8, 10;

  15, 12, 13, 14, 11;

  16, 20, 18, 19, 17, 21;

  28, 23, 26, 25, 24, 27, 22;

  ...

Row number k contains permutation of the k numbers:

{ (k^2-k+2)/2, (k^2-k+2)/2 + 1, ..., (k^2+k-2)/2 + 1 }.

MATHEMATICA

a[n_] := Module[{t, i, j},

t = Floor[(Sqrt[8n-7]-1)/2];

i = n-t(t+1)/2;

j = (t^2+3t+4)/2-n;

((i+j-1)(i+j-2) + ((-1)^Max[i, j]+1)i - ((-1)^Max[i, j]-1)j)/2];

Array[a, 55] (* Jean-François Alcover, Jan 26 2019 *)

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

m=((i+j-1)*(i+j-2)+((-1)**max(i, j)+1)*i-((-1)**max(i, j)-1)*j)/2

CROSSREFS

Cf. A056011, A056023, A057027, A064578, A194981, A194982, inverse functions A208233, A208234, A194280, A064790.

Sequence in context: A089852 A122308 A122307 * A305418 A284459 A106451

Adjacent sequences:  A188565 A188566 A188567 * A188569 A188570 A188571

KEYWORD

nonn

AUTHOR

Boris Putievskiy, Dec 27 2012

STATUS

approved

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Last modified August 15 12:29 EDT 2020. Contains 336497 sequences. (Running on oeis4.)