A177894 Determinant of the square matrix whose rows are the cyclic permutations of the digits of n.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, -3, -8, -15, -24, -35, -48, -63, -80, 4, 3, 0, -5, -12, -21, -32, -45, -60, -77, 9, 8, 5, 0, -7, -16, -27, -40, -55, -72, 16, 15, 12, 7, 0, -9, -20, -33, -48, -65, 25, 24, 21, 16, 9, 0, -11, -24, -39, -56, 36, 35, 32, 27, 20, 11, 0, -13, -28, -45, 49, 48, 45, 40, 33, 24, 13, 0, -15, -32, 64, 63, 60, 55, 48, 39, 28, 15, 0, -17, 81, 80, 77, 72, 65, 56, 45, 32

Offset: 0

Author

Michel Lagneau, Dec 15 2010

Keywords

• sign
• base
• easy
• look

Examples

			for n=104, the (3 X 3) matrix M is
  [1 0 4]
  [0 4 1]
  [4 1 0]
and a(104) = det(M) = -65.
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..9999

Crossrefs

Coincides with A257587 for the first 100 terms, but differs thereafter.

Programs

Mathematica

A177894[n_] := If[n < 10, n, Det[NestList[RotateLeft, IntegerDigits[n], IntegerLength[n]-1]]]; Array[A177894, 100, 0] (* Paolo Xausa, Mar 11 2024 *)

PARI

a(n) = {if(n<10, return(n)); my(d = digits(n), m, s); d = concat(d, d); s = #d/2; m = matrix(s, s, i, j, d[i+j-1]); matdet(m)} \\ David A. Corneth, Jun 12 2017

Sage

def A177894(n):
    d = n.digits()[::-1] if n > 0 else [0]
    M = Matrix(lambda i,j: d[(i+j) % len(d)], nrows=len(d))
    return M.determinant() # D. S. McNeil, Dec 16 2010

Formula

For n = a, det(M) = a;

for n = ab, det(M) = a^2 - b^2;

for n = abc, det(M) = 3abc - a^3 - b^3 - c^3; ...