Searching for Lynnes-stype Periodic Ratioanl Recurrences inspired ny Evgeni Lozitsky By Shalosh B. Ekhad In the delightful paper: "Fractional Linear Periodic Recurrences of orders t\ wo and three", by Evgeni Lozitsky he came up with several rational peridoic recurrencs of orders 2 and 3 with \ periods up to 12. Here we emulate him and reproduce, in Maple, all his recurrences and try to find new ones, by ex\ tending the search for all recurrences of the form z[n]=(1+a1*z[n-1]+...+ak*z[n-k])/z[n-k] for k from 2 to, 3, with periods from 2 to, 18 -------------------------------- Searching for recurrences of order, 2, of the form a[1] z[n - 1] + a[2] z[n - 2] + 1 z[n] = --------------------------------- z[n - 2] with periods up to R --------------------------------------- There exist periodic Lynnes-style recurrences of order, 2, and period, 5, here they are -z[n - 1] + 1 z[n] = ------------- z[n - 2] z[n - 1] + 1 z[n] = ------------ z[n - 2] and in Maple notation z[n] = (-z[n-1]+1)/z[n-2] z[n] = (z[n-1]+1)/z[n-2] -------------------------------- There exist periodic Lynnes-style recurrences of order, 2, and period, 8, here they are z[n] = ( 2 2 2 RootOf(_Z - RootOf(_Z + 2 _Z + 2)) (RootOf(_Z + 2 _Z + 2) + 1) z[n - 1] 2 2 + RootOf(_Z - RootOf(_Z + 2 _Z + 2)) z[n - 2] + 1)/z[n - 2] and in Maple notation z[n] = (RootOf(_Z^2-RootOf(_Z^2+2*_Z+2))*(RootOf(_Z^2+2*_Z+2)+1)*z[n-1]+RootOf( _Z^2-RootOf(_Z^2+2*_Z+2))*z[n-2]+1)/z[n-2] -------------------------------- There exist periodic Lynnes-style recurrences of order, 2, and period, 12, here they are 2 z[n] = (RootOf(_Z - %2) (2 %2 %1 + 8 %2 + %1 + 4) z[n - 1] 2 + RootOf(_Z - %2) z[n - 2] + 1)/z[n - 2] 2 %1 := RootOf(_Z + 4 _Z + 1) 2 %2 := RootOf(_Z + _Z - %1) and in Maple notation z[n] = (RootOf(_Z^2-RootOf(_Z^2+_Z-RootOf(_Z^2+4*_Z+1)))*(2*RootOf(_Z^2+_Z- RootOf(_Z^2+4*_Z+1))*RootOf(_Z^2+4*_Z+1)+8*RootOf(_Z^2+_Z-RootOf(_Z^2+4*_Z+1))+ RootOf(_Z^2+4*_Z+1)+4)*z[n-1]+RootOf(_Z^2-RootOf(_Z^2+_Z-RootOf(_Z^2+4*_Z+1)))* z[n-2]+1)/z[n-2] -------------------------------- There exist periodic Lynnes-style recurrences of order, 2, and period, 18, here they are 2 z[n] = (1/19 RootOf(_Z - %1) 5 4 3 2 (43 %1 + 114 %1 + 220 %1 + 273 %1 + 438 %1 + 122) z[n - 1] 2 + RootOf(_Z - %1) z[n - 2] + 1)/z[n - 2] 6 5 4 3 2 %1 := RootOf(_Z + 3 _Z + 6 _Z + 8 _Z + 12 _Z + 6 _Z + 1) and in Maple notation z[n] = (1/19*RootOf(_Z^2-RootOf(_Z^6+3*_Z^5+6*_Z^4+8*_Z^3+12*_Z^2+6*_Z+1))*(43* RootOf(_Z^6+3*_Z^5+6*_Z^4+8*_Z^3+12*_Z^2+6*_Z+1)^5+114*RootOf(_Z^6+3*_Z^5+6*_Z^ 4+8*_Z^3+12*_Z^2+6*_Z+1)^4+220*RootOf(_Z^6+3*_Z^5+6*_Z^4+8*_Z^3+12*_Z^2+6*_Z+1) ^3+273*RootOf(_Z^6+3*_Z^5+6*_Z^4+8*_Z^3+12*_Z^2+6*_Z+1)^2+438*RootOf(_Z^6+3*_Z^ 5+6*_Z^4+8*_Z^3+12*_Z^2+6*_Z+1)+122)*z[n-1]+RootOf(_Z^2-RootOf(_Z^6+3*_Z^5+6*_Z ^4+8*_Z^3+12*_Z^2+6*_Z+1))*z[n-2]+1)/z[n-2] -------------------------------- Searching for recurrences of order, 3, of the form a[1] z[n - 1] + a[2] z[n - 2] + a[3] z[n - 3] + 1 z[n] = ------------------------------------------------- z[n - 3] with periods up to R -------------------------------- There exist periodic Lynnes-style recurrences of order, 3, and period, 8, here they are -z[n - 2] - z[n - 1] + 1 z[n] = ------------------------ z[n - 3] z[n - 2] + z[n - 1] + 1 z[n] = ----------------------- z[n - 3] 2 2 -RootOf(_Z + 1) z[n - 1] + RootOf(_Z + 1) z[n - 2] + 1 z[n] = -------------------------------------------------------- z[n - 3] and in Maple notation z[n] = (-z[n-2]-z[n-1]+1)/z[n-3] z[n] = (z[n-2]+z[n-1]+1)/z[n-3] z[n] = (-RootOf(_Z^2+1)*z[n-1]+RootOf(_Z^2+1)*z[n-2]+1)/z[n-3] -------------------------------- There exist periodic Lynnes-style recurrences of order, 3, and period, 12, here they are 2 z[n] = (-z[n - 1] + (-1 + RootOf(_Z - _Z + 1)) z[n - 2] 2 + RootOf(_Z - _Z + 1) z[n - 3] + 1)/z[n - 3] 2 z[n] = (z[n - 1] + (1 + RootOf(_Z + _Z + 1)) z[n - 2] 2 + RootOf(_Z + _Z + 1) z[n - 3] + 1)/z[n - 3] 2 2 z[n] = (RootOf(_Z + 2) z[n - 1] - RootOf(_Z + 2) z[n - 2] 2 + RootOf(_Z + 2) z[n - 3] + 1)/z[n - 3] and in Maple notation z[n] = (-z[n-1]+(-1+RootOf(_Z^2-_Z+1))*z[n-2]+RootOf(_Z^2-_Z+1)*z[n-3]+1)/z[n-3 ] z[n] = (z[n-1]+(1+RootOf(_Z^2+_Z+1))*z[n-2]+RootOf(_Z^2+_Z+1)*z[n-3]+1)/z[n-3] z[n] = (RootOf(_Z^2+2)*z[n-1]-RootOf(_Z^2+2)*z[n-2]+RootOf(_Z^2+2)*z[n-3]+1)/z[ n-3] ------------------------------------- This ends this article that took, 25.423, seconds to generate