A389503 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 an n X n Toeplitz matrix whose first row consists of successive positive integer numbers 1, ..., n and whose first column consists of 1, n + 1, ..., 2*n - 1.

0, 0, 0, 38, 465, 11624, 207999, 5277958, 126680181, 3713452504, 111994892051, 3846931633078, 139521107394905, 5562408480673000, 235424229194638487, 10739841793463776838, 519216252222470801501, 26732328511050573296664, 1452812625138514950148795, 83408498684837216333348662

Offset

0,4

Comments

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

Links

Table of n, a(n) for n=0..19.

Wikipedia, Rule of Sarrus.

Example

a(3) = A322908(3) = 38:
  [1, 2, 3]
  [4, 1, 2]
  [5, 4, 1]

Mathematica

p[i_]:=i; M[i_, j_, n_]:=Part[Part[ToeplitzMatrix[Join[{p[1]}, Array[p, n-1, {n+1, 2*n-1}]], Array[p, n]], i], j]; 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, 20, 0]

Crossrefs

Cf. A322908, A389261, A389504.

Sequence in context: A240258 A254471 A039685 * A006418 A160317 A088891

Adjacent sequences: A389500 A389501 A389502 * A389504 A389505 A389506

Keyword

nonn

Author

Stefano Spezia, Oct 07 2025

Status

approved