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

1, -5, 78, -1200, 19680, -351360, 6854400, -145797120, 3367526400, -84072038400, 2258332876800, -64990937088000, 1995834890649600, -65167516237824000, 2254974602969088000, -82443156980760576000, 3176032637949050880000, -128603097714237898752000, 5460911310769351557120000

Offset

1,2

Comments

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

Links

Stefano Spezia, Table of n, a(n) for n = 1..400

Nicolay Avilov, Illustration of a(1)-a(5)

Formula

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

Example

         |  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 |

Mathematica

a={}; For[n=1, n<=19, n++, k=i=j=1; M[i, j]=k++; For[h=1, h<n, h++,  If[i==j==1, M[i, ++j]=k++; For[c=1, c<=h, c++, M[++i, j]=k++; M[i, --j]=k++], If[j==1 && i!=1, M[++i, j]=k++; For[c=1, c<=h, c++, M[i, ++j]=k++]; For[c=1, c<=h, c++, M[--i, j]=k++], If[i==1 &&j!=1, M[i, ++j]=k++; For[c=1, c<=h, c++, M[++i, j]=k++]; For[c=1, c<=h, c++, M[i, --j]=k++]]]]]; AppendTo[a, Det[Table[M[i, j], {i, n}, {j, n}]]]]; a (* Stefano 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 *)

Crossrefs

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

Sequence in context: A327903 A214443 A015973 * A136578 A057186 A106939

Adjacent sequences: A363373 A363374 A363375 * A363377 A363378 A363379

Keyword

sign

Author

Nicolay Avilov, May 29 2023

Extensions

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

Status

approved