A210521 -id:A210521 - cp's OEIS frontend

A199855 Inverse permutation to A210521.

1, 4, 2, 5, 3, 6, 11, 7, 12, 8, 13, 9, 14, 10, 15, 22, 16, 23, 17, 24, 18, 25, 19, 26, 20, 27, 21, 28, 37, 29, 38, 30, 39, 31, 40, 32, 41, 33, 42, 34, 43, 35, 44, 36, 45, 56, 46, 57, 47, 58, 48, 59, 49, 60, 50, 61, 51, 62, 52, 63, 53, 64, 54, 65, 55, 66, 79

Offset: 1

Boris Putievskiy, Feb 04 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.
Enumeration table T(n,k). The order of the list:
  T(1,1)=1;
  T(2,1), T(2,2), T(1,2), T(1,3), T(3,1),
  ...
  T(2,n-1), T(4,n-3), T(6,n-5), ..., T(n,1),
  T(2,n),   T(4,n-2), T(6,n-4), ..., T(n,2),
  T(1,n),   T(3,n-2), T(5,n-4), ..., T(n-1,2),
  T(1,n+1), T(3,n-1), T(5,n-3), ..., T(n+1,1),
  ...
The order of the list elements of adjacent antidiagonals. Let m be a positive integer.
Movement by antidiagonal {T(1,2*m), T(2*m,1)}     from T(2,2*m-1) to T(2*m,1)   length of step is 2,
movement by antidiagonal {T(1,2*m+1), T(2*m+1,1)} from T(2,2*m)   to T(2*m,2)   length of step is 2,
movement by antidiagonal {T(1,2*m), T(2*m,1)}     from T(1,2*m)   to T(2*m-1,2) length of step is 2,
movement by antidiagonal {T(1,2*m+1), T(2*m+1,1)} from T(1,2*m+1) to T(2*m+1,1) length of step is 2.
Table contains:
  row  1 is alternation of elements A001844 and A084849,
  row  2 is alternation of elements A130883 and A058331,
  row  3 is alternation of elements A051890 and A096376,
  row  4 is alternation of elements A033816 and A005893,
  row  6 is alternation of elements A100037 and A093328;
  row  5  accommodates elements A097080 in odd places,
  row  7  accommodates elements A137882 in odd places,
  row 10  accommodates elements A100038 in odd places,
  row 14  accommodates elements A100039 in odd places;
  column 1 is A093005 and alternation of elements A000384 and A001105,
  column 2 is alternation of elements A046092 and A014105,
  column 3 is A105638 and alternation of elements A014106 and A056220,
  column 4 is alternation of elements A142463 and A014107,
  column 5 is alternation of elements A091823 and A054000,
  column 6 is alternation of elements A090288 and |A168244|,
  column 8 is alternation of elements A059993 and A033537;
  column  7 accommodates elements  A071355  in odd  places,
  column  9 accommodates elements |A147973| in even places,
  column 10 accommodates elements  A139570  in odd  places,
  column 13 accommodates elements  A130861  in odd  places.

			The start of the sequence as table:
   1,  4,  5,  11,  13,  22,  25,  37,  41,  56,  61, ...
   2,  3,  7,   9,  16,  19,  29,  33,  46,  51,  67, ...
   6, 12, 14,  23,  26,  38,  42,  57,  62,  80,  86, ...
   8, 10, 17,  20,  30,  34,  47,  52,  68,  74,  93, ...
  15, 24, 27,  39,  43,  58,  63,  81,  87, 108, 115, ...
  18, 21, 31,  35,  48,  53,  69,  75,  94, 101. 123, ...
  28, 40, 44,  59,  64,  82,  88, 109, 116, 140, 148, ...
  32, 36, 49,  54,  70,  76,  95, 102, 124, 132, 157, ...
  45, 60, 65,  83,  89, 110, 117, 141, 149, 176, 185, ...
  50, 55, 71,  77,  96, 103, 125, 133, 158, 167, 195, ...
  66, 84, 90, 111, 118, 142, 150, 177, 186, 216, 226, ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   4,  2;
   5,  3,  6;
  11,  7, 12,  8;
  13,  9, 14, 10, 15;
  22, 16, 23, 17, 24, 18;
  25, 19, 26, 20, 27, 21, 28;
  37, 29, 38, 30, 39, 31, 40, 32;
  41, 33, 42, 34, 43, 35, 44, 36, 45;
  56, 46, 57, 47, 58, 48, 59, 49, 60, 50;
  61, 51, 62, 52, 63, 53, 64, 54, 65, 55, 66;
  ...
The start of the sequence as array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from the row number 2*r-2 of  triangle array, located above.
Last  2*r-1 numbers are from the row number 2*r-1 of  triangle array, located above.
   1;
   4, 2, 5, 3, 6;
  11, 7,12, 8,13, 9,14,10,15;
  22,16,23,17,24,18,25,19,26,20,27,21,28;
  37,29,38,30,39,31,40,32,41,33,42,34,43,35,44,36,45;
  56,46,57,47,58,48,59,49,60,50,61,51,62,52,63,53,64,54,65,55,66;
  ...
Row number r contains permutation numbers 4*r-3 from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-3*r+2,2*r*r-5*r+4, 2*r*r-3*r+3, 2*r*r-5*r+5, 2*r*r-3*r+4, 2*r*r-5*r+6, ..., 2*r*r-3*r+1, 2*r*r-r.
...

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. A210521, A001844, A084849, A130883, A058331, A051890, A096376, A033816, A005893, A100037, A093328, A097080, A137882, A100038, A100039, A093005, A000384, A001105, A046092, A014105, A105638, A014106, A056220, A142463, A014107, A091823, A054000, A090288, A168244, A059993, A033537, A071355, A147973, A139570, A130861.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=(2*j**2+(4*i-5)*j+2*i**2-3*i+2+(2+(-1)**j)*((1-(t+1)*(-1)**i)))/4

T(n,k) = (2*k^2+(4*n-5)*k+2*n^2-3*n+2+(2+(-1)^k)*((1-(k+n-1)*(-1)^i)))/4.

a(n) = (2*j^2+(4*i-5)*j+2*i^2-3*i+2+(2+(-1)^j)*((1-(t+1)*(-1)^i)))/4, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((sqrt(8*n-7) - 1)/2).

A216248 T(n,k)=((n+k)^2-4k+3-(-1)^k-(n+k-2)(-1)^(n+k))/2-1, if k=1 and (n mod 2)=1; T(n,k)=((n+k)^2-4k+3-(-1)^k-(n+k-2)(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n, k > 0.

Original entry on oeis.org

1, 2, 5, 3, 4, 6, 7, 10, 11, 14, 8, 9, 12, 13, 15, 16, 19, 20, 23, 24, 27, 17, 18, 21, 22, 25, 26, 28, 29, 32, 33, 36, 37, 40, 41, 44, 30, 31, 34, 35, 38, 39, 42, 43, 45, 46, 49, 50, 53, 54, 57, 58, 61, 62, 65, 47, 48, 51, 52, 55, 56, 59, 60, 63, 64, 66, 67

Offset: 1

Boris Putievskiy, Mar 14 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.
Enumeration table T(n,k). Let m be natural number. The order of the list:
T(1,1)=1;
T(1,2), T(1,3), T(2,2), T(2,1), T(3,1);
. . .
T(1,2*m), T(1,2*m+1), T(2,2*m), T(2,2*m-1), T(3,2*m-2), ... T(2*m-1,2), T(2*m-1,3), T(2*m,2), T(2*m,1), T(2*m+1,1);
. . .
Movement along two adjacent antidiagonals - step to the east, step to the southwest, step to the west, step to the southwest and so on. The length of each step is 1.

			The start of the sequence as table:
1....2...3...7...8..16..17...
5....4..10...9..19..18..32...
6...11..12..20..21..33..34...
14..13..23..22..36..35..53...
15..24..25..37..38..54..55...
27..26..40..39..57..56..78...
28..41..42..58..59..79..80...
. . .
The start of the sequence as triangular array read by rows:
1;
2,5;
3,4,6;
7,10,11,14;
8,9,12,13,15;
16,19,20,23,24,27;
17,18,21,22,25,26,28;
. . .
The start of the sequence as array read by rows, the length of row number r is 4*r-3.
First 2*r-2 numbers are from the row number 2*r-2 of triangle array, located above.
Last  2*r-1 numbers are from the row number 2*r-1 of triangle array, located above.
1;
2,5,3,4,6;
7,10,11,14,8,9,12,13,15;
16,19,20,23,24,27,17,18,21,22,25,26,28;
. . .
Row number r contains permutation of the 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-5*r+4, 2*r*r-5*r+7, ... 2*r*r-r-2, 2*r*r-r.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO]
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A213205, A213171, A213197, A210521; table T(n,k) contains: in rows A033816, A130883, A100037, A100038, A100039; in columns A000384, A071355, A014106, A091823, A130861.

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

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

A216249 T(n,k) = ((n+k)^2-4k+3-2(-1)^n+(-1)^k-(n+k-4)(-1)^(n+k))/2-2, if k=1 and (n mod 2)=1; T(n,k)=((n+k)^2-4k+3-2(-1)^n+(-1)^k-(n+k-4)(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n , k > 0.

Original entry on oeis.org

1, 3, 2, 4, 5, 6, 8, 7, 12, 11, 9, 10, 13, 14, 15, 17, 16, 21, 20, 25, 24, 18, 19, 22, 23, 26, 27, 28, 30, 29, 34, 33, 38, 37, 42, 41, 31, 32, 35, 36, 39, 40, 43, 44, 45, 47, 46, 51, 50, 55, 54, 59, 58, 63, 62, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 66, 68, 67, 72, 71, 76, 75, 80, 79, 84, 83, 88, 87

Offset: 1

Boris Putievskiy, Mar 14 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.
Enumeration table T(n,k). Let m be natural number. The order of the list:
T(1,1)=1;
T(2,1), T(1,2), T(1,3), T(2,2), T(3,1);
. . .
T(2,2*m-1), T(1,2*m), T(1,2*m+1), T(2,2*m), T(2*m-3,4), ... T(2*m,1), T(2*m-1,2), T(2*m-1,3), T(2*m,2), T(2*m+1,1);
. . .
Movement along two adjacent antidiagonals - step to the northeast, step to the east, step to the southwest, 3 steps to the west, 2 steps to the south and so on.
The length of each step is 1.

			The start of the sequence as table:
   1   3  4    8   9  17  18...
   2   5  7   10  16  19  29...
   6  12  13  21  22  34  35...
  11  14  20  23  33  36  50...
  15  25  26  38  39  55  56...
  24  27  37  40  54  57  75...
  28  42  43  59  60  80  81...
  ...
The start of the sequence as triangular array read by rows:
   1;
   3,  2;
   4,  5,  6;
   8,  7, 12, 11;
   9, 10, 13, 14, 15;
  17, 16, 21, 20, 25, 24;
  18, 19, 22, 23, 26, 27, 28;
  ...
As an array read by rows, where the length of row number r is 4*r-3:
First 2*r-2 numbers are from the row number 2*r-2 of triangle array, located above.
Last  2*r-1 numbers are from the row number 2*r-1 of triangle array, located above.
  1;
  3,   2,   4,   5,   6;
  8,   7,  12,  11,   9,  10,  13,  14,  15;
  17, 16,  21,  20,  25,  24,  18,  19,  22,  23,  26,  27,  28;
  ...
Row number r contains permutation of the 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-5*r+5, 2*r*r-5*r+4, ...2*r*r-r-1, 2*r*r-r.

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. A213205, A213171, A213197, A210521; table T(n,k) contains: in rows A100037, A033816, A130883, A100039, A100038; in columns A000384, A071355, A091823, A014106.

T[n_, k_] := ((n+k)^2 - 4k + 3 - 2(-1)^n + (-1)^k - (n+k-4)(-1)^(n+k))/2 - 2Boole[k == 1 && OddQ[n]];
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 20 2019 *)

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

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

A216250 T(n,k) = ((n+k)^2-4k+3-2(-1)^n-(-1)^k-(n+k-4)(-1)^(n+k))/2-3, if k=1 and (n mod 2)=1; T(n,k) = ((n+k)^2-4k+3-2(-1)^n-(-1)^k-(n+k-4)(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n, k > 0.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 7, 8, 11, 12, 10, 9, 14, 13, 15, 16, 17, 20, 21, 24, 25, 19, 18, 23, 22, 27, 26, 28, 29, 30, 33, 34, 37, 38, 41, 42, 32, 31, 36, 35, 40, 39, 44, 43, 45, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 49, 48, 53, 52, 57, 56, 61, 60, 65, 64, 66, 67

Offset: 1

Boris Putievskiy, Mar 14 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.
Enumeration table T(n,k). Let m be natural number. The order of the list:
  T(1,1)=1;
  T(1,2), T(2,1), T(2,2), T(1,3), T(3,1);
  . . .
  T(1,2*m), T(2,2*m-1), T(2,2*m), T(1,2*m+1), T(3,2*m-2), ... T(2*m-1,2), T(2*m,1), T(2*m,2), T(2*m-1,3), T(2*m+1,1);
  . . .
Movement along two adjacent antidiagonals - step to the southwest, step east, step to the northeast, 3 steps to the west, 2 steps to the south and so on. The length of each step is 1.

			The start of the sequence as table:
  1....2...5...7..10..16..19...
  3....4...8...9..17..18..30...
  6...11..14..20..23..33..36...
  12..13..21..22..34..35..51...
  15..24..27..37..40..54..57...
  25..26..38..39..55..56..76...
  28..41..44..58..61..79..82...
  . . .
The start of the sequence as triangular array read by rows:
  1;
  2,3;
  5,4,6;
  7,8,11,12;
  10,9,14,13,15;
  16,17,20,21,24,25;
  19,18,23,22,27,26,28;
  . . .
The start of the sequence as array read by rows, with length of row r: 4*r-3:
First 2*r-2 numbers are from the row number 2*r-2 of above triangle array.
Last  2*r-1 numbers are from the row number 2*r-1 of above triangle array.
  1;
  2,3,5,4,6;
  7,8,11,12,10,9,14,13,15;
  16,17,20,21,24,25,19,18,23,22,27,26,28;
  . . .
Row number r contains permutation of the 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r: 2*r*r-5*r+4, 2*r*r-5*r+5, ...2*r*r-r-2, 2*r*r-r.

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. A213205, A213171, A213197, A210521; table T(n,k) contains: in rows A130883, A033816, A100037, A100038, A100039; in columns A000384, A014106, A071355, A091823, A130861.

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

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

cp's OEIS Frontend

A199855 Inverse permutation to A210521.

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A216248 T(n,k)=((n+k)^2-4*k+3-(-1)^k-(n+k-2)*(-1)^(n+k))/2-1, if k=1 and (n mod 2)=1; T(n,k)=((n+k)^2-4*k+3-(-1)^k-(n+k-2)*(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n, k > 0.

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Formula

A216249 T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n+(-1)^k-(n+k-4)*(-1)^(n+k))/2-2, if k=1 and (n mod 2)=1; T(n,k)=((n+k)^2-4*k+3-2*(-1)^n+(-1)^k-(n+k-4)*(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n , k > 0.

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Programs

Mathematica

Python

Formula

A216250 T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n-(-1)^k-(n+k-4)*(-1)^(n+k))/2-3, if k=1 and (n mod 2)=1; T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n-(-1)^k-(n+k-4)*(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n, k > 0.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

A216248 T(n,k)=((n+k)^2-4k+3-(-1)^k-(n+k-2)(-1)^(n+k))/2-1, if k=1 and (n mod 2)=1; T(n,k)=((n+k)^2-4k+3-(-1)^k-(n+k-2)(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n, k > 0.

A216249 T(n,k) = ((n+k)^2-4k+3-2(-1)^n+(-1)^k-(n+k-4)(-1)^(n+k))/2-2, if k=1 and (n mod 2)=1; T(n,k)=((n+k)^2-4k+3-2(-1)^n+(-1)^k-(n+k-4)(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n , k > 0.

A216250 T(n,k) = ((n+k)^2-4k+3-2(-1)^n-(-1)^k-(n+k-4)(-1)^(n+k))/2-3, if k=1 and (n mod 2)=1; T(n,k) = ((n+k)^2-4k+3-2(-1)^n-(-1)^k-(n+k-4)(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n, k > 0.