A216252 A213196 as table read layer by layer clockwise.

1, 4, 5, 2, 3, 7, 10, 8, 6, 11, 9, 17, 20, 23, 14, 12, 13, 16, 26, 38, 43, 39, 21, 24, 15, 22, 25, 30, 42, 58, 63, 48, 35, 31, 27, 18, 19, 29, 34, 57, 53, 69, 76, 70, 64, 49, 36, 32, 28, 37, 33, 47, 52, 81, 75, 95, 102, 109, 88, 82, 54, 59, 44, 40, 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).

The order of the list:

T(1,1)=1;

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

. . .

T(1,n), T(2,n), ... T(n-1,n), T(n,n), T(n,n-1), ... T(n,1);

. . .

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, MathWorld: 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=min(t; n-(t-1)^2), j=min(t; t^2-n+1), 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;
  4,5,2;
  3,7,10,8,6;
  11,9,17,20,23,14,12;
  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=min(t, n-(t-1)**2)
j=min(t, t**2-n+1)
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, A211377, A214928, A060734, A060736.

Sequence in context: A344531 A213171 A261098 * A335615 A328622 A338248

Adjacent sequences: A216249 A216250 A216251 * A216253 A216254 A216255

Keyword

nonn,tabl

Author

Boris Putievskiy, Mar 15 2013

Status

approved