A209293 -id:A209293 - cp's OEIS frontend

A210535 Second inverse function (numbers of columns) for pairing function A209293.

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

Offset: 1

Boris Putievskiy, Jan 28 2013

nonn

			The start of the sequence as triangle array read by rows:
  1;
  2,1;
  2,3,1;
  2,4,3,1;
  2,4,5,3,1;
  2,4,6,5,3,1;
  2,4,6,7,5,3,1;
  2,4,6,8,7,5,3,1;
  . . .
Row number r contains permutation numbers from 1 to r: 2,4,6,...2*floor(r/2),2*floor(r/2)-1,2*floor(r/2)-3,...3,1.

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

Cf. A209293, A200260, A101688, A003056, A220073.

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

a(n) = 2*A200260(n)-A101688(n)*(4*A002260(n)-2*A003056(n)-3).

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

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.

A220073 Mirror of the triangle A130517.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Dec 03 2012

T(n,k) = A130517(n,n-k+1), 1 <= k <= n;
T(n,n) = T(n,1) + 1.
From Boris Putievskiy, Jan 15 2013: (Start)
General case see A187760. Let m be natural number. Table T(n,k) n, k > 0, T(n,k)=n-k+1, if n>=k, T(n,k)=k-n+m-1, if n < k.  Table T(n,k) read by antidiagonals. The first column of the table T(n,1) is the sequence of the natural numbers A000027. In all columns with number k (k > 1) the segment with the length of (k-1): {m+k-2, m+k-3, ..., m}  shifts the sequence A000027. For m=1 the result is A220073, for m=2 the result is A143182. (End)
First inverse function (numbers of rows) for pairing function A209293. - Boris Putievskiy, Jan 28 2013

			From _Boris Putievskiy_, Jan 15 2013: (Start)
The start of the sequence as table:
1..1..2..3..4..5..6..7...
2..1..1..2..3..4..5..6...
3..2..1..1..2..3..4..5...
4..3..2..1..1..2..3..4...
5..4..3..2..1..1..2..3...
6..5..4..3..2..1..1..2...
7..6..5..4..3..2..1..1...
8..7..6..5..4..3..2..1...
. . .
The start of the sequence as triangle array read by rows:
1,
1, 2,
2, 1, 3,
3, 1, 2, 4,
4, 2, 1, 3, 5,
5, 3, 1, 2, 4, 6,
6, 4, 2, 1, 3, 5, 7,
7, 5, 3, 1, 2, 4, 6, 8,
. . .
Row number r contains r numbers: r-1, r-3,...,1,...r-2,r.
(End)

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A028310 (left edge), A000027 (right edge), A000012 (central terms), A000217 (row sums), A220075 (partial sums in rows), A002260, A000027, A143182, A187760, A209293.

a220073 n k = a220073_tabl !! (n-1) !! (k-1)
a220073_row n = a220073_tabl !! (n-1)
a220073_tabl = map reverse a130517_tabl

max = 13;
row[n_] := Join[Range[n, 1, -1], Range[max - n + 1]];
T = Array[row, max];
Table[T[[n - k + 1, k]], {n, 1, max}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 11 2017 *)

T(1,1)=1, for n>1: T(n,k)=T(n-1,n-k+1), 1<=k

From Boris Putievskiy, Jan 15 2013: (Start)

For the general case

a(n) = |(t+1)^2 - 2n| + m*floor((t^2+3t+2-2n)/(t+1)),

where t = floor((-1+sqrt(8*n-7))/2).

For m = 2

a(n) = |(t+1)^2 - 2n| + floor((t^2+3t+2-2n)/(t+1)),

where t=floor((-1+sqrt(8*n-7))/2). (End)

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A210535 Second inverse function (numbers of columns) for pairing function A209293.

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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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A220073 Mirror of the triangle A130517.

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