A220603 -id:A220603 - cp's OEIS frontend

A081344 Natural numbers in square maze arrangement, read by antidiagonals.

1, 2, 4, 9, 3, 5, 10, 8, 6, 16, 25, 11, 7, 15, 17, 26, 24, 12, 14, 18, 36, 49, 27, 23, 13, 19, 35, 37, 50, 48, 28, 22, 20, 34, 38, 64, 81, 51, 47, 29, 21, 33, 39, 63, 65, 82, 80, 52, 46, 30, 32, 40, 62, 66, 100, 121, 83, 79, 53, 45, 31, 41, 61, 67, 99, 101, 122, 120, 84, 78, 54

Offset: 1

Paul Barry, Mar 19 2003

Arrange the natural numbers by taking clockwise and counterclockwise turns. Begin (LL) and then repeat (RRR)(LLL).
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. - Boris Putievskiy, Dec 16 2012
For generalizations see A219159, A213928. - Boris Putievskiy, Mar 10 2013

			The start of the sequence as table T(i,j), i,j > 0:
   1   4    5    16 ...
   2   3    6    15 ...
   9   8    7    14 ...
  10  11   12    13 ...
  ....

Boris Putievskiy, Rows n = 1..100 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
Index entries for sequences that are permutations of the natural numbers

Cf. A219159, A213928. The main diagonal is A002061. The following appear within interlaced sequences: A016754, A001844, A053755, A004120. The first row is A081345. The first column is A081346. The inverse permutation A194280, the first inverse function (numbers of rows) A220603, the second inverse function (numbers of columns) A220604.

T[n_, k_] := T[n, k] = Which[OddQ[n] && k==1, n^2, EvenQ[k] && n==1, k^2, EvenQ[n] && k==1, T[n-1, 1]+1, OddQ[k] && n==1, T[1, k-1]+1, k <= n, T[n, k-1]+1 - 2 Mod[n, 2], True, T[n-1, k]-1 + 2 Mod[k, 2]]; Table[T[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 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
if j >= i:
     m=(j-1)**2 + j + (j-i)*(-1)**(j-1)
else:
     m=(i-1)**2 + i - (i-j)*(-1)**(i-1)
# Boris Putievskiy, Dec 19 2012

from math import isqrt
def A081344(n):
    t = (k:=isqrt(m:=n<<1))+((m<<2)>(k<<2)*(k+1)+1)-1
    i, j = n-(t*(t+1)>>1), (t*(t+3)>>1)+2-n
    r = max(i,j)
    return (r-1)**2+r+(j-i if r&1 else i-j) # Chai Wah Wu, Nov 04 2024

From Boris Putievskiy, Dec 19 2012: (Start)
a(n) = (i-1)^2 + i + (i-j)*(-1)^(i-1) if i >= j,
a(n) = (j-1)^2 + j - (j-i)*(-1)^(j-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). (End)
Enumeration by boustrophedonic ("ox-plowing") method: If i >= j: T(i,j)=(i-1)^2+i + (i-j)*(-1)^(i-1), if i  < j: T(i,j)=(j-1)^2+j - (j-i)*(-1)^(j-1). - Boris Putievskiy, Dec 19 2012
T(i,j) = m^2 - m + 1 - (i - j)*(-1)^m where m = max(i,j). - Ziad Ahmed, Jun 09 2025

A194195 First inverse function (numbers of rows) for pairing function A060734.

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Dec 21 2012

The sequence is the second inverse function (numbers of columns) for pairing function A060736.

			The start of the sequence as triangle array read by rows:
1;
2,2,1;
3,3,3,2,1;
4,4,4,4,3,2,1;
. . .
Row number k contains 2k-1 numbers k,k,...k,k-1,k-2,...1 (k times repetition "k").

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A060734, A060736, A220603, A220604

f[n_]:=Module[{t=Floor[Sqrt[n-1]]+1},Min[t,t^2-n+1]]; Array[f,80] (* Harvey P. Dale, Dec 31 2012 *)

t=int(math.sqrt(n-1)) +1
i=min(t,t**2-n+1)

a(n) = min{t; t^2 - n + 1}, where t=floor(sqrt(n-1))+1.

A194258 Second inverse function (numbers of columns) for pairing function A060734.

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Dec 21 2012

The sequence is the first inverse function (numbers of rows) for pairing function A060736.

			The start of the sequence as triangle array read by rows:
1;
1,2,2;
1,2,3,3,3;
1,2,3,4,4,4,4;
. . .
Row number k contains 2k-1 numbers 1,2,...k-1,k,k,...k (k times repetition "k").

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A060734, A060736, A220603, A220604.

Flatten[Table[Join[Range[n-1],Table[n,{n}]],{n,10}]] (* Harvey P. Dale, Jun 23 2013 *)

t=int(math.sqrt(n-1)) +1
j=min(t,n-(t-1)**2)

a(n) = min{t; n - (t - 1)^2}, where 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

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

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A081344 Natural numbers in square maze arrangement, read by antidiagonals.

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A194195 First inverse function (numbers of rows) for pairing function A060734.

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A194258 Second inverse function (numbers of columns) for pairing function A060734.

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

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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.

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