A185180 -id:A185180 - cp's OEIS frontend

A209278 Second inverse function (numbers of rows) for pairing function A185180.

1, 2, 1, 2, 3, 1, 3, 2, 4, 1, 3, 4, 2, 5, 1, 4, 3, 5, 2, 6, 1, 4, 5, 3, 6, 2, 7, 1, 5, 4, 6, 3, 7, 2, 8, 1, 5, 6, 4, 7, 3, 8, 2, 9, 1, 6, 5, 7, 4, 8, 3, 9, 2, 10, 1, 6, 7, 5, 8, 4, 9, 3, 10, 2, 11, 1

Offset: 1

Boris Putievskiy, Jan 15 2013

			The start of the sequence as table T(r,s) r,s >0 read by antidiagonals:
1...2...2...3...3...4...4...5...
1...3...2...4...3...5...4...6...
1...4...2...5...3...6...4...7...
1...5...2...6...3...7...4...8...
1...6...2...7...3...8...4...9...
1...7...2...8...3...9...4..10...
1...8...2...9...3..10...4..11...
. . .
The start of the sequence as triangle array read by rows:
1;
2, 1;
2, 3, 1;
3, 2, 4, 1;
3, 4, 2, 5, 1;
4, 3, 5, 2, 6, 1;
4, 5, 3, 6, 2, 7, 1;
5, 4, 6, 3, 7, 2, 8, 1;
. . .
Row number r contains permutation numbers form 1 to r.
If r is odd (r+1)/2, (r+1)/2 +1, (r+1)/2 -1, ... 2, r, 1.
If r is even r/2 + 1, r/2, r/2 + 2, ...  2, r, 1.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions

Cf. A185180, A092542, A092543.

T[r_, s_] := If[OddQ[s], (s+1)/2, r + s/2];
Table[T[r-s+1, s], {r, 1, 11}, {s, r, 1, -1}] // Flatten (* Jean-François Alcover, Nov 19 2019 *)

T(r,s)=s\2+if(bittest(s,0),1,r) \\ - M. F. Hasler, Jan 15 2013

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

a(n) = floor((A003056(n)+3)/2) + floor(A002260(n)/2)*(-1)^(A002260(n)+A003056(n)).
a(n)= floor((t+3)/2)+ floor(i/2)*(-1)^(i+t),
where t=floor((-1+sqrt(8*n-7))/2), i=n-t*(t+1)/2.
T(r,2s-1)=s, T(r,2s)= r+s. (When read as square array by antidiagonals.)

A209279 First inverse function (numbers of rows) for pairing function A185180.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 2, 3, 1, 4, 3, 2, 4, 1, 5, 3, 4, 2, 5, 1, 6, 4, 3, 5, 2, 6, 1, 7, 4, 5, 3, 6, 2, 7, 1, 8, 5, 4, 6, 3, 7, 2, 8, 1, 9, 5, 6, 4, 7, 3, 8, 2, 9, 1, 10, 6, 5, 7, 4, 8, 3, 9, 2, 10, 1, 11, 6, 7, 5, 8, 4, 9, 3, 10, 2, 11, 1, 12, 7, 6, 8, 5, 9, 4, 10, 3, 11, 2, 12, 1, 13

Offset: 1

Boris Putievskiy, Jan 15 2013

The triangle equals A158946 with the first column removed. - Georg Fischer, Jul 26 2023

			The start of the sequence as table T(r,s) r,s >0 read by antidiagonals:
  1...1...2...2...3...3...4...4...
  2...1...3...2...4...3...5...4...
  3...1...4...2...5...3...6...4...
  4...1...5...2...6...3...7...4...
  5...1...6...2...7...3...8...4...
  6...1...7...2...8...3...9...4...
  7...1...8...2...9...3..10...4...
  ...
The start of the sequence as triangle array read by rows:
  1;
  1, 2;
  2, 1, 3;
  2, 3, 1, 4;
  3, 2, 4, 1, 5;
  3, 4, 2, 5, 1, 6;
  4, 3, 5, 2, 6, 1, 7;
  4, 5, 3, 6, 2, 7, 1, 8;
  ...
Row number r contains permutation numbers form 1 to r.
If r is odd (r+1)/2, (r+1)/2-1, (r+1)/2+1,...r-1, 1, r.
If r is even r/2, r/2+1, r/2-1, ... r-1, 1, r.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions

Cf. A158946, A185180, A128180, A092542, A092543, A209278.

T[n_, k_] := Abs[(2*k - 1 + (-1)^(n - k)*(2*n + 1))/4];
Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd *)

T(n, k)=abs((2*k-1+(-1)^(n-k)*(2*n+1))/4) \\ Andrew Howroyd, Dec 31 2017

# Edited by M. F. Hasler, May 30 2020
def a(n):
   t = int((math.sqrt(8*n-7) - 1)/2);
   i = n-t*(t+1)/2;
   return int(t/2)+1+int(i/2)*(-1)**(i+t+1)

a(n) = floor((A003056(n)+2)/2)+ floor(A002260(n)/2)*(-1)^(A002260(n)+A003056(n)+1).
a(n) = |A128180(n)|.
a(n) = floor((t+2)/2) + floor(i/2)*(-1)^(i+t+1), where t=floor((-1+sqrt(8*n-7))/2), i=n-t*(t+1)/2.
T(r,2s)=s, T(r,2s-1)= r+s-1.(When read as table T(r,s) by antidiagonals.)
T(n,k) = ceiling((n + (-1)^(n-k)*k)/2) = (n+k)/2 if n-k even, otherwise (n-k+1)/2. - M. F. Hasler, May 30 2020

Data corrected by Andrew Howroyd, Dec 31 2017

A209293 Inverse permutation of A185180.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 8, 9, 7, 10, 13, 12, 14, 11, 15, 18, 19, 17, 20, 16, 21, 25, 24, 26, 23, 27, 22, 28, 32, 33, 31, 34, 30, 35, 29, 36, 41, 40, 42, 39, 43, 38, 44, 37, 45, 50, 51, 49, 52, 48, 53, 47, 54, 46, 55, 61, 60, 62, 59, 63, 58, 64, 57, 65, 56, 66, 72, 73, 71, 74, 70, 75, 69, 76, 68, 77, 67

Offset: 1

Boris Putievskiy, Jan 16 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) by diagonals. The order of the list
if n is odd  - T(n-1,2),T(n-3,4),...,T(2,n-1),T(1,n),T(3,n-2),...T(n,1).
if n is even - T(n-1,2),T(n-3,4),...,T(3,n-2),T(1,n),T(2,n-1),...T(n,1).
Table T(n,k) contains:
Column number 1 A000217,
column number 2 A000124,
column number 3 A000096,
column number 4 A152948,
column number 5 A034856,
column number 6 A152950,
column number 7 A055998.
Row    number 1 A000982,
row    number 2 A097063.

			The start of the sequence as table:
  1....2...5...8..13..18...25...32...41...
  3....4...9..12..19..24...33...40...51...
  6....7..14..17..26..31...42...49...62...
  10..11..20..23..34..39...52...59...74...
  15..16..27..30..43..48...63...70...87...
  21..22..35..38..53..58...75...82..101...
  28..29..44..47..64..69...88...95..116...
  36..37..54..57..76..81..102..109..132...
  45..46..65..68..89..94..117..124..149...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  2,3;
  5,4,6;
  8,9,7,10;
  13,12,14,11,15;
  18,19,17,20,16,21;
  25,24,26,23,27,22,28;
  32,33,31,34,30,35,29,36;
  41,40,42,39,43,38,44,37,45;
  . . .
Row number r contains permutation from r numbers:
if r is odd  ceiling(r^2/2), ceiling(r^2/2)+1, ceiling(r^2/2)-1, ceiling(r^2/2)+2, ceiling(r^2/2)-2,...r*(r+1)/2;
if r is even ceiling(r^2/2), ceiling(r^2/2)-1, ceiling(r^2/2)+1, ceiling(r^2/2)-2, ceiling(r^2/2)+2,...r*(r+1)/2;

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. A000217, A000124, A000096, A152948, A034856, A152950, A055998, A000982, A097063.

max = 10; row[n_] := Table[Ceiling[(n + k - 1)^2/2] + If[OddQ[k], 1, -1]*Floor[n/2], {k, 1, max}]; t = Table[row[n], {n, 1, max}]; Table[t[[n - k + 1, k]], {n, 1, max}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 17 2013 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
m1=int((i+j)/2)+int(i/2)*(-1)**(i+t+1)
m2=int((i+j+1)/2)+int(i/2)*(-1)**(i+t)
m=(m1+m2-1)*(m1+m2-2)/2+m1

As table T(n,k) read by antidiagonals
T(n,k) = n*n/2+4*(floor((k-1)/2)+1)*n+ceiling((k-1)^2/2), n,k > 0.
As linear sequence
a(n) = (m1+m2-1)*(m1+m2-2)/2+m1, where
m1 = int((i+j)/2)+int(i/2)*(-1)^(i+t+1),
m2 = int((i+j+1)/2)+int(i/2)*(-1)^(i+t),
t = int((math.sqrt(8*n-7) - 1)/ 2),
i = n-t*(t+1)/2,
j = (t*t+3*t+4)/2-n.

A214928 A209293 as table read layer by layer clockwise.

Original entry on oeis.org

1, 2, 4, 3, 5, 9, 14, 7, 6, 8, 12, 17, 23, 20, 11, 10, 13, 19, 26, 34, 43, 30, 27, 16, 15, 18, 24, 31, 39, 48, 58, 53, 38, 35, 22, 21, 25, 33, 42, 52, 63, 75, 88, 69, 64, 47, 44, 29, 28, 32, 40, 49, 59, 70, 82, 95, 109, 102, 81, 76, 57, 54, 37, 36, 41, 51, 62

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). 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....2...5...8..13..18...
  3....4...9..12..19..24...
  6....7..14..17..26..31...
  10..11..20..23..34..39...
  15..16..27..30..43..48...
  21..22..35..38..53..58...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  2,4,3;
  5,9,14,7,6;
  8,12,17,23,20,11,10;
  13,19,26,34,43,30,27,16,15;
  18,24,31,39,48,58,53,38,35,22,21;
  . . .
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. A209293, A209279, A209278, A185180, A060734, A060736; table T(n,k) contains: in rows A000982, A097063; in columns A000217, A000124, A000096, A152948, A034856, A152950, A055998, A000982, A097063.

t=int((math.sqrt(n-1)))+1
i=min(t,n-(t-1)**2)
j=min(t,t**2-n+1)
m1=int((i+j)/2)+int(i/2)*(-1)**(2*i+j-1)
m2=int((i+j+1)/2)+int(i/2)*(-1)**(2*i+j-2)
result=(m1+m2-1)*(m1+m2-2)/2+m1

As table
T(n,k) = n*n/2+4*(floor((k-1)/2)+1)*n+ceiling((k-1)^2/2), n,k > 0.
As linear sequence
a(n)= (m1+m2-1)*(m1+m2-2)/2+m1, where m1=floor((i+j)/2) + floor(i/2)*(-1)^(2*i+j-1), m2=int((i+j+1)/2)+int(i/2)*(-1)^(2*i+j-2), where i=min(t; n-(t-1)^2), j=min(t; t^2-n+1), t=floor(sqrt(n-1))+1.

A214929 A209293 as table read layer by layer - layer clockwise, layer counterclockwise and so on.

Original entry on oeis.org

1, 3, 4, 2, 5, 9, 14, 7, 6, 10, 11, 20, 23, 17, 12, 8, 13, 19, 26, 34, 43, 30, 27, 16, 15, 21, 22, 35, 38, 53, 58, 48, 39, 31, 24, 18, 25, 33, 42, 52, 63, 75, 88, 69, 64, 47, 44, 29, 28, 36, 37, 54, 57, 76, 81, 102, 109, 95, 82, 70, 59, 49, 40, 32, 41, 51, 62

Offset: 1

Boris Putievskiy, Mar 11 2013

nonn

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). 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....2...5...8..13..18...
  3....4...9..12..19..24...
  6....7..14..17..26..31...
  10..11..20..23..34..39...
  15..16..27..30..43..48...
  21..22..35..38..53..58...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  3,4,2;
  5,9,14,7,6;
  10,11,20,23,17,12,8;
  13,19,26,34,43,30,27,16,15;
  21,22,35,38,53,58,48,39,31,24,18;
  . . .
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. A081344, A209293, A209279, A209278, A185180; table T(n,k) contains: in rows A000982, A097063; in columns A000217, A000124, A000096, A152948, A034856, A152950, A055998, A000982, A097063.

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=int((i+j)/2)+int(i/2)*(-1)**(2*i+j-1)
m2=int((i+j+1)/2)+int(i/2)*(-1)**(2*i+j-2)
result=(m1+m2-1)*(m1+m2-2)/2+m1

As table
T(n,k) = n*n/2+4*(floor((k-1)/2)+1)*n+ceiling((k-1)^2/2), n,k > 0.
As linear sequence
a(n)= (m1+m2-1)*(m1+m2-2)/2+m1, where
m1=floor((i+j)/2) + floor(i/2)*(-1)^(2*i+j-1), m2=int((i+j+1)/2)+int(i/2)*(-1)^(2*i+j-2),
where 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

A209278 Second inverse function (numbers of rows) for pairing function A185180.

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A209279 First inverse function (numbers of rows) for pairing function A185180.

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A209293 Inverse permutation of A185180.

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A214928 A209293 as table read layer by layer clockwise.

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A214929 A209293 as table read layer by layer - layer clockwise, layer counterclockwise and so on.

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