A081344 -id:A081344 - cp's OEIS frontend

0, 1, 3, 8, 2, 4, 9, 7, 5, 15, 24, 10, 6, 14, 16, 25, 23, 11, 13, 17, 35, 48, 26, 22, 12, 18, 34, 36, 49, 47, 27, 21, 19, 33, 37, 63, 80, 50, 46, 28, 20, 32, 38, 62, 64, 81, 79, 51, 45, 29, 31, 39, 61, 65, 99, 120, 82, 78, 52, 44, 30, 40, 60, 66, 98, 100

Offset: 0

Omar E. Pol, Feb 09 2013

This sequence consists of 0 together with a permutation of the natural numbers. The structure is the same as A081344 but starting with 0, not 1.

It appears that in the n-th layer there is at least a prime number <= g and also there is at least a prime number > g, where g is the number on the main diagonal, the n-th oblong number A002378(n), if n >= 1.

			The first layer is [1, 2, 3] which looks like this:
.  3,
1, 2,
The second layer is [4, 5, 6, 7, 8] which looks like this:
.  .  4
.  .  5,
8, 7, 6,
Square array T(0,0)..T(10,10) begins:
0,     3,   4,  15,  16,  35,  36,  63,  64,  99, 100,...
1,     2,   5,  14,  17,  34,  37,  62,  65,  98, 101,...
8,     7,   6,  13,  18,  33,  38,  61,  66,  97, 102,...
9,    10,  11,  12,  19,  32,  39,  60,  67,  96, 103,...
24,   23,  22,  21,  20,  31,  40,  59,  68,  95, 104,...
25,   26,  27,  28,  29,  30,  41,  58,  69,  94, 105,...
48,   47,  46,  45,  44,  43,  42,  57,  70,  93, 106,...
49,   50,  51,  52,  53,  54,  55,  56,  71,  92, 107,...
80,   79,  78,  77,  76,  75,  74,  73,  72,  91, 108,...
81,   82,  83,  84,  85,  86,  87,  88,  89,  90, 109,...
120, 119, 118, 117, 116, 115, 114, 113, 112, 111, 110,...
...

Index entries for sequences that are permutations of the nonnegative integers

Main diagonal is A002378.

Cf. A000040, A000290, A002620, A081344, A220508.

a(n) = A081344(n+1) - 1.
T(n,k) = n^2 + k     , if n is odd  and k<=n.
T(n,k) = n(n + 2) - k, if n is even and k<=n.
T(n,k) = k(k + 2) - n, if n is odd  and nT(n,k) = k^2 + n     , if n is even and n

A093650 Natural numbers arranged in a square maze beginning 1, 2, 3, then moving right, then up, right, down, left, down, right, etc., and read by antidiagonals upwards.

Original entry on oeis.org

1, 2, 6, 3, 5, 7, 12, 4, 8, 20, 13, 11, 9, 19, 21, 30, 14, 10, 18, 22, 42, 31, 29, 15, 17, 23, 41, 43, 56, 32, 28, 16, 24, 40, 44, 72, 57, 55, 33, 27, 25, 39, 45, 71, 73, 90, 58, 54, 34, 26, 38, 46, 70, 74, 110, 91, 89, 59, 53, 35, 37, 47, 69, 75, 109, 111

Offset: 1

Michael Joseph Halm, May 15 2004

			a(3) = 6 because the maze begins 2 under 1, 3 under 2, 4 right of 3, 5 right of 2 and 6 right of 1.
Array begins:
   1   6---7  20 ...
   |   |   |   |
   2   5   8  19 ...
   |   |   |   |
   3---4   9  18 ...
           |   |
  12--11--10  17 ...
   |           |
  13--14--15--16 ...
  ...

Cf. A080335, A081347, A081348.

Other square mazes: A081344, A081349.

More terms from Jinyuan Wang, Jun 15 2022

A213928 Natural numbers placed in table T(n,k) layer by layer. The order of placement - at the beginning 2 layers counterclockwise, next 1 layer clockwise and so on. T(n,k) read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Mar 06 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.In general, let b(z) be a sequence of integer 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). Natural numbers placed in table T(n,k) layer by layer. The order of placement - layer is counterclockwise, if b(z) is odd; layer is clockwise if b(z) is even. T(n,k) read by antidiagonals.For A219159 - the order of the placement - at the beginning m layers counterclockwise, next m layers clockwise and so on - b(z)=floor((z-1)/m)+1. For this sequence b(z)=z^2 mod 3.

			The start of the sequence as table.
The direction of the placement denotes by ">" and  "v".
  ..........v...........v...........v
  >1....4...5..16..25..26..49..64..65...
  >2....3...6..15..24..27..48..63..66...
  .9....8...7..14..23..28..47..62..67...
  >10..11..12..13..22..29..46..61..68...
  >17..18..19..20..21..30..45..60..69...
  .36..35..34..33..32..31..44..59..70...
  >37..38..39..40..41..42..43..58..71...
  >50..51..52..53..54..55..56..57..72...
  .81..80..79..78..77..76..75..74..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  4,2;
  5,3,9;
  16,6,8,10;
  25,15,7,11,17;
  26,24,14,12,18,36;
  49,27,23,13,19,35,37;
  64,48,28,22,20,34,38,50;
  65,63,47,29,21,33,39,51,81;
  . . .

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. A219159, A081344, A194280, A042964, A130196, A011655, A220516.

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

For general case.
As table
T(n,k) = ((1+(-1)^(b(k)-1))*(k^2-n+1)-(-1+(-1)^(b(k)-1))*((k-1)^2 +n))/2, if  k >= n;
T(n,k) = ((1+(-1)^b(n))*(n^2-k+1)-(-1+(-1)^b(n))*((n-1)^2 +k))/2, if  n >k.
As linear sequence
a(n) = ((1+(-1)^(b(j)-1))*(j^2-i+1)-(-1+(-1)^(b(j)-1))*((j-1)^2 +i))/2, if  j >= i;
a(n) = ((1+(-1)^b(i))*(i^2-j+1)-(-1+(-1)^b(i))*((i-1)^2 +j))/2, 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).
For this sequence b(z)=z^2 mod 3.
As table
T(n,k) = ((1+(-1)^(k^2 mod 3-1))*(k^2-n+1)-(-1+(-1)^(k^2 mod 3-1))*((k-1)^2 +n))/2, if  k >= n;
T(n,k) = ((1+(-1)^(n^2 mod 3))*(n^2-k+1)-(-1+(-1)^(n^2 mod 3))*((n-1)^2 +k))/2, if  n >k.
As linear sequence
a(n) = ((1+(-1)^(j^2 mod 3-1))*(j^2-i+1)-(-1+(-1)^(j^2 mod 3-1))*((j-1)^2 +i))/2, if  j >= i;
a(n) = ((1+(-1)^(i^2 mod 3))*(i^2-j+1)-(-1+(-1)^(i^2 mod 3))*((i-1)^2 +j))/2, 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).

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.

A363376 Determinant 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, -5, 78, -1200, 19680, -351360, 6854400, -145797120, 3367526400, -84072038400, 2258332876800, -64990937088000, 1995834890649600, -65167516237824000, 2254974602969088000, -82443156980760576000, 3176032637949050880000, -128603097714237898752000, 5460911310769351557120000

Offset: 1

Nicolay Avilov, May 29 2023

sign

The matrix is the upper-left n X n part of the square arrangement in A081344.
Number i is in the matrix at row A220604(i) column A220603(i), for i = 1..n^2.
Conjecture: a(n) has trailing zeros for n > 3. - Stefano Spezia, May 31 2023
The conjecture is true and its proof follows easily from Detlef Meya's formula. - Stefano Spezia, Apr 20 2024

			         |  1----2    9---10   25 |
         |       |    |    |    | |
         |  4----3    8   11   24 |
         |  |         |    |    | |
  a(5) = |  5----6----7   12   23 | = 19680.
         |                 |    | |
         | 16---15---14---13   22 |
         |  |                   | |
         | 17---18---19---20---21 |

Stefano Spezia, Table of n, a(n) for n = 1..400
Nicolay Avilov, Illustration of a(1)-a(5)

Cf. A081344, A220603, A220604, A363460 (permanent).

a={}; For[n=1, n<=19, n++,k=i=j=1; M[i,j]=k++; For[h=1, hStefano Spezia, May 31 2023 *)
a={1};For[n=2,n<20,n++,AppendTo[a,(-1)^(n-1)*2^(n-3)*(2*n*(n-1)+1)*n!]];a (* Detlef Meya, Jun 11 2023 *)

a(1) = 1, for a > 1: a(n) = (-1)^(n-1)*2^(n-3)*(2*n*(n-1)+1)*(n!). - Detlef Meya, Jun 11 2023

E.g.f.: x*(2 + 7*x + 20*x^2 + 12*x^3)/(2*(1 + 2*x)^3). - Stefano Spezia, Apr 20 2024

a(16)-a(19) from Stefano Spezia, May 31 2023

A216253 A213196 as table read layer by layer - layer clockwise, layer counterclockwise and so on.

Original entry on oeis.org

1, 2, 5, 4, 3, 7, 10, 8, 6, 12, 14, 23, 20, 17, 9, 11, 13, 16, 26, 38, 43, 39, 21, 24, 15, 18, 27, 31, 35, 48, 63, 58, 42, 30, 25, 22, 19, 29, 34, 57, 53, 69, 76, 70, 64, 49, 36, 32, 28, 40, 44, 59, 54, 82, 88, 109, 102, 95, 75, 81, 52, 47, 33, 37, 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). 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....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;
  2,5,4;
  3,7,10,8,6;
  12,14,23,20,17,9,11;
  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, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A213196, A081344, A211377, A214929.

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=(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=(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.

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

Previous Showing 11-17 of 17 results.

cp's OEIS Frontend

A220516 Nonnegative integers in square maze arrangement T(n,k), read by antidiagonals, n>=0, k>=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A093650 Natural numbers arranged in a square maze beginning 1, 2, 3, then moving right, then up, right, down, left, down, right, etc., and read by antidiagonals upwards.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A213928 Natural numbers placed in table T(n,k) layer by layer. The order of placement - at the beginning 2 layers counterclockwise, next 1 layer clockwise and so on. T(n,k) read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

A363376 Determinant of the n X n matrix formed by placing 1..n^2 in L-shaped gnomons in alternating directions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A216253 A213196 as table read layer by layer - layer clockwise, layer counterclockwise and so on.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica