A185725 -id:A185725 - cp's OEIS frontend

A060734 Natural numbers written as a square array ending in last row from left to right and rightmost column from bottom to top are read by antidiagonals downwards.

1, 4, 2, 9, 3, 5, 16, 8, 6, 10, 25, 15, 7, 11, 17, 36, 24, 14, 12, 18, 26, 49, 35, 23, 13, 19, 27, 37, 64, 48, 34, 22, 20, 28, 38, 50, 81, 63, 47, 33, 21, 29, 39, 51, 65, 100, 80, 62, 46, 32, 30, 40, 52, 66, 82, 121, 99, 79, 61, 45, 31, 41, 53, 67, 83, 101

Offset: 1

Frank Ellermann, Apr 23 2001

A simple permutation of natural numbers.
Parity of the sequence is given by A057211 (n-th run has length n). - Jeremy Gardiner, Dec 26 2008
The square with corners T(1,1)=1 and T(n,n)=n^2-n+1 is occupied by the numbers 1,2,...,n^2. - Clark Kimberling, Feb 01 2011
a(n) is pairing function - function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} - the set of integer positive numbers. - Boris Putievskiy, Dec 17 2012

			Northwest corner:
.1  4  9 16 ..  => a(1) =  1
.2  3  8 15 ..  => a(2) =  4, a(3) = 2
.5  6  7 14 ..  => a(4) =  9, a(5) = 3, a(6) = 5
10 11 12 13 ..  => a(7) = 16, a(8) = 8, a(9) = 6, a(10)=10

Alois P. Heinz, Rows n = 1..141 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060736. Inverse: A064790.

Cf. A185725, A185726, A185728.

T:= (n,k)-> `if`(n<=k, k^2-n+1, (n-1)^2+k):
seq(seq(T(n, d-n), n=1..d-1), d=2..15);

f[n_, k_]:=k^2-n+1/; k>=n;
f[n_, k_]:=(n-1)^2+k/; kClark Kimberling, Feb 01 2011 *)

T(n,k) = (n-1)^2+k, T(k, n)=n^2+1-k, 1 <= k <= n.
From Clark Kimberling, Feb 01 2011: (Start)
T(1,k) = k^2 (A000290).
T(n,n) = n^2-n+1 (A002061).
T(n,1) = (n-1)^2+1 (A002522). (End)

Corrected by Jeremy Gardiner, Dec 26 2008

A185728 Array associated with squares, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 8, 10, 6, 7, 15, 17, 11, 9, 14, 24, 26, 18, 12, 13, 23, 35, 37, 27, 19, 16, 22, 34, 48, 50, 38, 28, 20, 21, 33, 47, 63, 65, 51, 39, 29, 25, 32, 46, 62, 80, 82, 66, 52, 40, 30, 31, 45, 61, 79, 99, 101, 83, 67, 53, 41, 36, 44, 60, 78, 98, 120, 122, 102, 84, 68, 54, 42, 43, 59, 77, 97, 119, 143, 145, 123, 103, 85, 69, 55, 49, 58, 76, 96, 118, 142, 168, 170, 146, 124, 104, 86, 70, 56, 57, 75, 95, 117, 141, 167, 195

Offset: 1

Clark Kimberling, Feb 01 2011

Every positive integer occurs exactly once; hence, as a sequence, A185725 is a permutation of the positive integers.  The square with corners T(0,0)=1 and T(n,n)=n^2 is occupied by the numbers 1,2,...,n^2.
T(1,k)=(k-1)^2+1 (A002522)
T(n,1)=-1+n^2 for n>=2.

			Northwest corner:
1...2...5...10...17
3...4...6...11...18
8...7...9...12...19
15..14..13..16...20

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A185725, A185726, A060734.

f[n_,k_]:=(k-1)^2+n/; k>n;
f[n_,n_]:=n^2; f[n_,k_]:=n^2-k/; k

T(n,k)=(k-1)^2+n if nk.

A185726 Array associated with squares, by antidiagonals.

Original entry on oeis.org

1, 3, 4, 8, 10, 10, 18, 22, 24, 21, 35, 44, 45, 48, 39, 61, 80, 81, 84, 86, 66, 98, 134, 138, 136, 144, 142, 104, 148, 210, 222, 216, 220, 231, 220, 155, 213, 312, 339, 332, 325, 340, 351, 324, 221, 295, 444, 495, 492, 475, 480, 504, 510, 458, 304, 396, 610, 696, 704, 680, 666, 690, 720, 714, 626, 406, 518, 814, 948, 976, 950, 918, 924, 965, 996, 969, 832, 529, 663, 1060, 1257, 1316, 1295, 1248, 1225, 1260, 1315, 1340, 1281, 1080, 675, 833, 1352, 1629, 1732, 1725

Offset: 1

Clark Kimberling, Feb 01 2011

Every positive integer occurs exactly once; hence, as a sequence, A185725 is a permutation of the positive integers.  The square with corners T(0,0)=1 and T(n,n)=n^2 is occupied by the numbers 1,2,...,n^2.
T(n,1)=n^2 (A000290)
T(n,n)=(n-1)^2+1 (A002522)
T(1,k)=k^2-1 (A132411).

			Northwest corner:
1...3...8...15...24
4...2...6...13...22
9...7...5...11...20
16..14..12..10...18

Cf. A185725, A060734, A185728.

f[n_,k_]:=n^2-2*k+2/; n>=k;
f[n_,k_]:=k^2-2*n+1/; n

T(n,k)=n^2-2k+2 if n>=k; T(n,k)=k^2-2n+1 if n

A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 9, 8, 13, 17, 16, 6, 14, 21, 26, 25, 11, 12, 22, 31, 37, 36, 18, 15, 20, 32, 43, 50, 49, 27, 24, 23, 30, 44, 57, 65, 64, 38, 35, 19, 33, 42, 58, 73, 82, 81, 51, 48, 28, 29, 45, 56, 74, 91, 101, 100, 66, 63, 39, 34, 41, 59, 72, 92, 111

Offset: 1

Boris Putievskiy, Mar 11 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.
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(1,2), T(2,1), T(2,2);
  . . .
  T(1,n), T(3,n), ... T(n,3), T(n,1); T(n,2), T(n,4), ... T(4,n), T(2,n);
  . . .

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

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 A002522, A000290, A059100, A005563, A117950, A008865, A087475, A028872, A117951, A028347, A114949, A028875, A117619, A028878, A189833, A028881, A189834, A028884, A114948, A028560, A189836; in columns A002061, A014206, A002378, A027688, A028387, A027689, A028552, A027690, A014209, A027691, A027692, A082111, A027693, A028557, A027694, A108195, A187710, A048058, A048840.

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*i-i+(j%2)*(2-(j+1)/2)+((j+1)%2)*(j/2+1)
else:
   result=j*j-2*(i%2)*j + (i%2)*((i+1)/2+1) + ((i+1)%2)*(-i/2+1)

As table
T(n,k) = k*k-2*(n mod 2)*k+(n mod 2)*((n+1)/2+1)+((n+1) mod 2)*(-n/2+1), if n<=k;
T(n,k) = n*n-n+(k mod 2)*(2-(k+1)/2)+((k+1) mod 2)*(k/2+1), if n>k.
As linear sequence
a(n) = j*j-2*(i mod 2)*j+(i mod 2)*((i+1)/2+1)+((i+1) mod 2)*(-i/2+1), if i<=j;
a(n) = i*i-i+(j mod 2)*(2-(j+1)/2)+((j+1) mod 2)*(j/2+1), 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).

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.

Original entry on oeis.org

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).

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A060734 Natural numbers written as a square array ending in last row from left to right and rightmost column from bottom to top are read by antidiagonals downwards.

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A185728 Array associated with squares, by antidiagonals.

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A185726 Array associated with squares, by antidiagonals.

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A214870 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

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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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