A note on the derived quotients of certain cyclically presented groups

We consider the derived quotients, An, of the cyclically presented groups G(n)= < x(i)|x(i+1)(-1)x(i+2)x(i+1)(-1) x(i+2)x(i)x(i+1)(-1)x(i) = 1 >(n) and show that A(n) is 2-generated, isomorphic to Zd(1) x Zd(2), d(2)|d(1) with order o(A(n)) = 2(n+1) (1 - cos n theta) where theta = arccos (sic). We also prove that o(A(n)) satisfies the recurrence relation a(n) = 5a(n-1) - 10a(n-2) + 8a(n-3).