1,2

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). Table read by boustrophedonic ("ox-plowing") method.

Let m be natural number. The order of the list:

T(1,1)=1;

T(2,1), T(2,2), T(1,2);

. . .

T(1,2*m+1), T(2,2*m+1), ... T(2*m,2*m+1), T(2*m+1,2*m+1), T(2*m+1,2*m), ... T(2*m+1,1);

T(2*m,1), T(2*m,2), ... T(2*m,2*m-1), T(2*m,2*m), T(2*m-1,2*m), ... T(1,2*m);

. . .

The first row is layer read clockwise, the second row is layer counterclockwise.

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 Weisstein's World of Mathematics, Pairing functions

a(n) = (m1+m2-1)*(m1+m2-2)/2+m1, where m1=(3*i+j-1-(-1)^i+(i+j-2)*(-1)^(i+j))/4, m2=((1+(-1)^i)*((1+(-1)^j)*2*int((j+2)/4)-(-1+(-1)^j)*(2*int((i+4)/4)+2*int(j/2)))-(-1+(-1)^i)*((1+(-1)^j)*(1+2*int(i/4)+2*int(j/2))-(-1+(-1)^j)*(1+2*int(j/4))))/4, i=(t mod 2)*min(t; n-(t-1)^2) + (t+1 mod 2)*min(t; t^2-n+1), j=(t mod 2)*min(t; t^2-n+1) + (t+1 mod 2)*min(t; n-(t-1)^2), t=floor(sqrt(n-1))+1.

The start of the sequence as table:

1....4...3..11..13...

2....5...7...9..16...

6....8..10..17..26...

12..14..23..20..38...

15..24..21..39..43...

. . .

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

1;

2,5,4;

3,7,10,8,6;

12,14,23,20,17,9,11;

13,16,26,38,43,39,21,24,15;

. . .

Row number r contains 2*r-1 numbers.

(Python)

t=int((math.sqrt(n-1)))+1

i=(t % 2)*min(t, n-(t-1)**2) + ((t+1) % 2)*min(t, t**2-n+1)

j=(t % 2)*min(t, t**2-n+1) + ((t+1) % 2)*min(t, n-(t-1)**2)

m1=(3*i+j-1-(-1)**i+(i+j-2)*(-1)**(i+j))/4

m2=((1+(-1)**i)*((1+(-1)**j)*2*int((j+2)/4)-(-1+(-1)**j)*(2*int((i+4)/4)+2*int(j/2)))-(-1+(-1)**i)*((1+(-1)**j)*(1+2*int(i/4)+2*int(j/2))-(-1+(-1)**j)*(1+2*int(j/4))))/4

m=(m1+m2-1)*(m1+m2-2)/2+m1

Cf. A213196, A081344, A211377, A214929.

Sequence in context: A361832 A361945 A171837 * A115303 A266403 A266415

Adjacent sequences: A216250 A216251 A216252 * A216254 A216255 A216256

nonn,tabl

Boris Putievskiy, Mar 15 2013

approved