A216253 A213196 as table read layer by layer - layer clockwise, layer counterclockwise and so on.

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, 18, 27, 31, 35, 48, 63, 58, 42, 30, 25, 22, 19, 29, 34, 57, 53, 69, 76, 70, 64, 49, 36, 32, 28, 40, 44, 59, 54, 82, 88, 109, 102, 95, 75, 81, 52, 47, 33, 37, 41, 46, 62

Offset

1,2

Comments

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.

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 Weisstein's World of Mathematics, Pairing functions

Index entries for sequences that are permutations of the natural numbers

Formula

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.

Example

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.

Prog

(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

Crossrefs

Cf. A213196, A081344, A211377, A214929.

Sequence in context: A361832 A361945 A171837 * A115303 A266403 A266415

Adjacent sequences: A216250 A216251 A216252 * A216254 A216255 A216256

Keyword

nonn,tabl

Author

Boris Putievskiy, Mar 15 2013

Status

approved