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
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
KEYWORD
nonn,tabl
AUTHOR
Boris Putievskiy, Dec 27 2012
STATUS
approved