A389427 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 M(n) formed by writing the numbers 1, ..., n^2 successively forward and backward along the rows in zig-zag pattern.

0, 0, 0, 0, 0, 0, -63936, -12009900, -3116605440, -827479103724, -260422616952000, -96512435591057640, -40164348509586604800, -20052269790848124630840, -10936415815917472073854080, -7113402179426306212616312100, -4970562554160989406882352594944, -4094443282484159068661708211382500

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(6) = -63936:
  [ 1,  2,  3,  4,  5,  6]
  [12, 11, 10,  9,  8,  7]
  [13, 14, 15, 16, 17, 18]
  [24, 23, 22, 21, 20, 19]
  [25, 26, 27, 28, 29, 30]
  [36, 35, 34, 33, 32, 31]

Mathematica

M[i_, j_, n_] := 1 - j + i n + (-1 + 2 j - n) Mod[i, 2]; 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. A322277, A389261, A389428.

Sequence in context: A119276 A145437 A236873 * A236608 A214758 A265827

Adjacent sequences: A389424 A389425 A389426 * A389428 A389429 A389430

Keyword

sign

Author

Stefano Spezia, Oct 03 2025

Status

approved