A389261 a(n) = Sum_{i=1..n} (Product_{j=1..n} M(j, ((i+j-2) mod n)+1) - Product_{j=1..n} M(j, ((i-j-1) mod n)+1)) where M is the n X n matrix with numbers 1, 2, ..., n^2 in order across rows.

0, 0, 0, 0, 0, 0, -3456, -1680700, -656015360, -257682513924, -109238438400000, -51314410315015848, -26971484454564249600, -15900584540593785814600, -10502594915812780982104064, -7751030669527270764328312500, -6368927485184314316526213660672, -5804243263631170614915627777847500

Offset

0,7

Comments

The definition generalizes the rule of Sarrus to matrices of order different than 3.

Links

Table of n, a(n) for n=0..17.

Wikipedia, Rule of Sarrus.

Example

a(1) = 1 - 1 = 0:
  [1]
a(2) = 1*4 + 2*3 - (2*3 + 1*4) = 0:
  [1, 2]
  [3, 4]
a(3) = det(M) = 1*5*9 + 2*6*7 + 3*4*8 - (3*5*7 + 1*6*8 + 2*4*9) = 0:
  [1, 2, 3]
  [4, 5, 6]
  [7, 8, 9]
a(6) = -3456:
  [ 1,  2,  3,  4,  5,  6]
  [ 7,  8,  9, 10, 11, 12]
  [13, 14, 15, 16, 17, 18]
  [19, 20, 21, 22, 23, 24]
  [25, 26, 27, 28, 29, 30]
  [31, 32, 33, 34, 35, 36]

Mathematica

M[i_, j_, n_]:=j+(i-1)n; a[n_]:=Sum[Product[M[j, Mod[i+j-2, n]+1, n], {j, n}]-Product[M[j, Mod[i-j-1, n]+1, n], {j, n}], {i, n}]; Array[a, 18, 0]

Crossrefs

Cf. A232773.

Sequence in context: A179705 A234893 A251485 * A082242 A321866 A235835

Adjacent sequences: A389258 A389259 A389260 * A389262 A389263 A389264

Keyword

sign

Author

Stefano Spezia, Sep 27 2025

Status

approved