1,2

Garrity, p. 218, states "Since the integers form an Abelian group, every subgroup including nZ, is normal and thus Z/nZ will form a group. It is common to represent each coset in Z/nZ by an integer between 0 and (n-1): Z/nZ = {0,1,2,...,(n-1}." Performing the analogous operation with the Z/3Z addition table (as a matrix), generates A007070: 1, 4, 14, 48, 164, 560... The recursion multipliers 4, 20, -32, -96 are present with changed signs in the characteristic polynomial of M: x^4 - 4x^3 - 20x^2 + 32x + 96.

Thomas A. Garrity, "All the Mathematics You Missed But Need to Know for Graduate School", Cambridge University Press, 2002.

Let M = the Z/4Z = {0, 1, 2, 3} addition table considered as a matrix = [0 1 2 3 / 1 2 3 0 / 2 3 0 1 / 3 0 1 2]. Then a(n) = 2nd term from left in M^n * [1 0 0 0].

The recursion operation (n>4) is a(n+4) = 4*a(n+3) + 20*a(n+2) - 32*a(n+1) - 96*a(n).

a(3) = 48 since M^3 * [1 0 0 0] = [44 48 60 64] (a(3) = 2nd term from left.

a(8) = 419840 = 4*69504 + 20*11648 - 32*1888 - 96*320.

a[n_] := (MatrixPower[{{0, 1, 2, 3}, {1, 2, 3, 0}, {2, 3, 0, 1}, {3, 0, 1, 2}}, n].{{1}, {0}, {0}, {0}})[[2, 1]]; Table[ a[n], {n, 22}] (from Robert G. Wilson v Jun 16 2004)

Cf. A007070.

Sequence in context: A051823 A037507 A037690 this_sequence A025013 A144014 A131681

Adjacent sequences: A095894 A095895 A095896 this_sequence A095898 A095899 A095900

nonn

Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 11 2004

Edited, corrected and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 16 2004