A060736 -id:A060736 - cp's OEIS frontend

A214871 Natural numbers placed in table T(n,k) layer by layer. The order of placement - T(n,n), T(1,n), T(n,1), T(2,n), T(n,2),...T(n-1,n), T(n,n-1). Table T(n,k) read by antidiagonals.

1, 3, 4, 6, 2, 7, 11, 8, 9, 12, 18, 13, 5, 14, 19, 27, 20, 15, 16, 21, 28, 38, 29, 22, 10, 23, 30, 39, 51, 40, 31, 24, 25, 32, 41, 52, 66, 53, 42, 33, 17, 34, 43, 54, 67, 83, 68, 55, 44, 35, 36, 45, 56, 69, 84, 102, 85, 70, 57, 46, 26, 47, 58, 71, 86, 103, 123

Offset: 1

Boris Putievskiy, Mar 11 2013

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.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1).
Enumeration table T(n,k) layer by layer. The order of the list:
  T(1,1)=1;
  T(2,2), T(1,2), T(2,1);
  . . .
  T(n,n), T(1,n), T(n,1), T(2,n), T(n,2),...T(n-1,n), T(n,n-1);
  . . .

			The start of the sequence as table:
  1....3...6..11..18..27...
  4....2...8..13..20..29...
  7....9...5..15..22..31...
  12..14..16..10..24..33...
  19..21..23..25..17..35...
  28..30..32..34..36..26...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  3,4;
  6,2,7;
  11,8,9,12;
  18,13,5,14,19;
  27,20,15,16,21,28;
  . . .

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. A060734, A060736, A185725, A213921, A213922; table T(n,k) contains: in rows A059100, A087475, A114949, A189833, A114948, A114962; in columns A117950, A117951, A117619, A189834, A189836; the main diagonal is A002522.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i == j:
   result=(i-1)**2+1
if i > j:
   result=(i-1)**2+2*j+1
if i < j:
   result=(j-1)**2+2*i

As table
T(n,k) = (n-1)^2+1,     if n=k;
T(n,k) = (n-1)^2+2*k+1, if n>k;
T(n,k) = (k-1)^2+2*n,   if nAs linear sequence
a(n) = (i-1)^2+1,     if i=j;
a(n) = (i-1)^2+2*j+1, if i>j;
a(n) = (j-1)^2+2*i,   if i>j; where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A216252 A213196 as table read layer by layer clockwise.

Original entry on oeis.org

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.

A363460 a(n) is the permanent of the n X n matrix formed by placing 1..n^2 in L-shaped gnomons in alternating directions.

Original entry on oeis.org

1, 1, 11, 556, 74964, 21700112, 11500685084, 10057140949968, 13496937368200000, 26331147893897760544, 71606290155732170272320, 262516365211410942628577408, 1262517559940020030446967822592, 7786463232979127181938238723356160, 60414239829783205320232261233394491136

Offset: 0

Stefano Spezia, Jun 03 2023

nonn

The matrix is the upper-left n X n part of the square arrangement in A081344.

The matrix element k is at row A220604(k) and column A220603(k), for k = 1..n^2.

			a(5) = 21700112 is the permanent of the 5 X 5 matrix
  |  1----2    9---10   25 |
  |       |    |    |    | |
  |  4----3    8   11   24 |
  |  |         |    |    | |
  |  5----6----7   12   23 |
  |                 |    | |
  | 16---15---14---13   22 |
  |  |                   | |
  | 17---18---19---20---21 |

Cf. A006527 (trace), A037270 (elements sum of the matrix), A060736, A061349 (anti trace), A081344, A220603, A220604, A363376 (determinant).

a={1}; For[n=1, n<=14, n++,k=i=j=1; M[i,j]=k++; For[h=1, h

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A214871 Natural numbers placed in table T(n,k) layer by layer. The order of placement - T(n,n), T(1,n), T(n,1), T(2,n), T(n,2),...T(n-1,n), T(n,n-1). Table T(n,k) read by antidiagonals.

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A216252 A213196 as table read layer by layer clockwise.

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A363460 a(n) is the permanent of the n X n matrix formed by placing 1..n^2 in L-shaped gnomons in alternating directions.

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