A213196 -id:A213196 - cp's OEIS frontend

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

Boris Putievskiy, Mar 15 2013

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);
  . . .

			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.

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

Cf. A213196, A211377, A214928, A060734, A060736.

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

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.

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

Original entry on oeis.org

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

Boris Putievskiy, Mar 15 2013

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.

			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.

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

Cf. A213196, A081344, A211377, A214929.

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

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.

cp's OEIS Frontend

A216252 A213196 as table read layer by layer clockwise.

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