initialized to 1,1,-1,x, Nest[Append[Rest[#], Factor[(#[[4]] #[[2]] + #[[3]]^2)/prnt@#[[1]]]] &, {1, 1, -1, x}, 17]; 1 1 -1 x 1+x -1-x+x^2 -1-x-x^3 -x (2+3 x) 1+3 x+3 x^2-x^4+x^5 -(1+x) (-1-2 x+2 x^2+3 x^3-x^4+x^5) -1-3 x-3 x^2-5 x^3-9 x^4-3 x^5+2 x^6-x^7 -x (-1-x+x^2) (3+9 x+9 x^2+5 x^3+2 x^4+x^6) 1+6 x+15 x^2+16 x^3-6 x^5+12 x^6+13 x^7+x^9+x^10 -(1+x+x^3) (1+5 x+4 x^2-15 x^3-30 x^4-13 x^5+6 x^6+x^7-3 x^8+x^9) -(1+x) (1+5 x+10 x^2+20 x^3+50 x^4+63 x^5+12 x^6-19 x^7+8 x^8+5 x^9-2 x^10+5 x^11-2 x^12+x^13) x (2+3 x) (-2-12 x-27 x^2-27 x^3-6 x^4+18 x^5+23 x^6+3 x^7-2 x^8+14 x^9+6 x^10-3 x^11+2 x^12) 1+10 x+45 x^2+110 x^3+135 x^4+45 x^5-2 x^6+211 x^7+429 x^8+311 x^9+105 x^10+69 x^11+56 x^12+5 x^13-9 x^14+6 x^15+4 x^16-3 x^17+x^18 . . . never manages to produce a denominator? (Very) similarly for 1,i,1,x: Nest[Append[Rest[#], Factor[(#[[4]] #[[2]] + #[[3]]^2)/prnt@#[[1]], Extension -> I]] &, {1, I, 1, x}, 17] 1 i 1 x 1+i x -i+x-i x^2 1+i x-i x^3 -i x (-2 i+3 x) 1+3 i x-3 x^2-x^4+i x^5 (1+i x) (i-2 x+2 i x^2-3 x^3+i x^4+x^5) 1+3 i x-3 x^2-5 i x^3+9 x^4+3 i x^5+2 x^6-i x^7 -x (1+i x+x^2) (-3-9 i x+9 x^2+5 i x^3-2 x^4+x^6) 1+6 i x-15 x^2-16 i x^3-6 i x^5-12 x^6-13 i x^7+i x^9-x^10 (1+i x-i x^3) (-i+5 x+4 i x^2+15 x^3+30 i x^4-13 x^5+6 i x^6-x^7+3 i x^8+x^9) -(-i+x) (-i+5 x+10 i x^2-20 x^3-50 i x^4+63 x^5+12 i x^6+19 x^7-8 i x^8+5 x^9-2 i x^10-5 x^11+2 i x^12+x^13) x (-2 i+3 x) (-2 i+12 x+27 i x^2-27 x^3-6 i x^4-18 x^5-23 i x^6+3 x^7-2 i x^8-14 x^9-6 i x^10-3 x^11+2 i x^12) 1+10 i x-45 x^2-110 i x^3+135 x^4+45 i x^5+2 x^6-211 i x^7+429 x^8+311 i x^9-105 x^10-69 i x^11+56 x^12+5 i x^13+9 x^14-6 i x^15+4 x^16-3 i x^17-x^18 --rwg