#############
#    r=1    #
#############

x^3+(2*x-1)*(x^2-4*x+1)*F+x^3*F^2 (=0)

#############
#    r=2    #
#############

144*x^3*(x+2)+(-168*x^4-840*x^3-744*x^2+336*x-24)*F-4*x*(3*x^4-10*x^3-97*x^2-146*x+1)*F^2+2*x^3*(x+4)*(11*x+23)*F^3+x^4*(x+4)^2*F^4  (=0)

#############
#    r=3    #
#############

16*(64*x^6-80*x^5+128*x^4-72*x^3+1248*x^2+166*x-1)*x^3*(4*x+3)*(64*x^7-12304*x^6+1168160*x^5+58072*x^4-1483056*x^3+201042*x^2+676863*x+11664)+(4194304*x^17-316735488*x^16-10919641088*x^15-11607773184*x^14-33879937024*x^13+43040624640*x^12-111269412864*x^11-336643549184*x^10+167590306816*x^9+420849985280*x^8-138443655552*x^7-347562926912*x^6-105480829952*x^5+5333115152*x^4+1652378400*x^3-40804560*x^2-830520*x+5832)*F+4*x*(7340032*x^16-106381312*x^15-948744192*x^14+5698564096*x^13+29244638208*x^12-33632006144*x^11-114402076416*x^10-42761499136*x^9+1080847424*x^8+5202950768*x^7+106577929632*x^6+134513565488*x^5+53164591120*x^4+5561083488*x^3+186295860*x^2+2536569*x-18225)*F^2+2*x^2*(58720256*x^15+581959680*x^14+8706158592*x^13+13878194176*x^12+11987657472*x^11+49028197632*x^10+137497145088*x^9+139816788992*x^8-88046980928*x^7-363777262144*x^6-374283577920*x^5-179941745712*x^4-38547786828*x^3-2419968708*x^2-37764441*x+171315)*F^3+x^3*(293601280*x^14+3937927168*x^13+6148583424*x^12-42046171136*x^11-181067272448*x^10+106616392192*x^9+949540282624*x^8+1112127624384*x^7+450056230976*x^6+125791307776*x^5+154302597760*x^4+84474468016*x^3+11518738668*x^2+301482216*x-661203)*F^4+8*x^4*(58720256*x^13+459800576*x^12-1132052480*x^11-7594454272*x^10-3972674176*x^9+17025258560*x^8+27432696352*x^7+17549449992*x^6+5305148280*x^5-2218829376*x^4-3820288471*x^3-1435607406*x^2-78629022*x+729)*F^5+4*x^5*(117440512*x^12+120586240*x^11-2920644608*x^10-7695873536*x^9+15579358720*x^8+68179659280*x^7+76221365584*x^6+27913506672*x^5-3164022160*x^4-2094940124*x^3+757631061*x^2+160784244*x+473850)*F^6+4*x^6*(67108864*x^11-222298112*x^10-1645084672*x^9-867139584*x^8+2710728640*x^7+2714202112*x^6+1254824368*x^5+2581628232*x^4+2498103550*x^3+636041502*x^2-46912365*x-531441)*F^7+2*x^7*(33554432*x^10-239075328*x^9-614858752*x^8+3253329920*x^7+11316039136*x^6+12249556904*x^5+4242483584*x^4-1084831864*x^3-895682934*x^2-97245684*x-492075)*F^8-8*x^8*(32*x+27)*(458752*x^7-22528*x^6+913728*x^5+3963198*x^4+2535675*x^3-773758*x^2-731241*x-13122)*F^9+4*x^9*(4096*x^5+45568*x^4+62508*x^3+17167*x^2-5013*x-243)*(32*x+27)^2*F^10-18*x^10*(32*x^2+31*x+3)*(32*x+27)^3*F^11+x^11*(32*x+27)^4*F^12 (=0)



###########################
#  Verbose version:  r=1  #  
###########################

`The Minimal Algebraic Equation Satisfied by the Generating Function Enumerating Words with exactly one pattern 123, and of length`, n
`in the alphabet {1, ...,n}, where Each Letter Appears Exactly`, 1, ` Times . `
`Theorem: Let a(n) be the number of words in the alphabet {1, ...,n} where each letter occurs exactly`, 1, `times `
`and that have exactly one pattern 123, in other words, there is exactly one i1<i2<i3 such that w[i1]<w[i2]<w[i3] `
`Let F be the ordinary generating function `
``
F = Sum(a(n)*x^n, n = 0 .. infinity)
`then F satisfies the following algebraic equation (derived rigorously via the Yang scheme and Groebner Bases)`
``
x^3+(2*x-1)*(x^2-4*x+1)*F+x^3*F^2 = 0
``
`This ends this article, that took`, 0.24e-1, `seconds to generate `


###########################
#  Verbose version:  r=2  #  
###########################

`The Minimal Algebraic Equation Satisfied by the Generating Function Enumerating Words with exactly one pattern 123, and of length`, 2*n
`in the alphabet {1, ...,n}, where Each Letter Appears Exactly`, 2, ` Times . `
`Theorem: Let a(n) be the number of words in the alphabet {1, ...,n} where each letter occurs exactly`, 2, `times `
`and that have exactly one pattern 123, in other words, there is exactly one i1<i2<i3 such that w[i1]<w[i2]<w[i3] `
`Let F be the ordinary generating function `
``
F = Sum(a(n)*x^n, n = 0 .. infinity)
`then F satisfies the following algebraic equation (derived rigorously via the Yang scheme and Groebner Bases)`
``
144*x^3*(x+2)+(-168*x^4-840*x^3-744*x^2+336*x-24)*F-4*x*(3*x^4-10*x^3-97*x^2-146*x+1)*F^2+2*x^3*(x+4)*(11*x+23)*F^3+x^4*(x+4)^2*F^4 = 0
``
`This ends this article, that took`, 0.1e-2, `seconds to generate `


###########################
#  Verbose version:  r=3  #  
###########################

`The Minimal Algebraic Equation Satisfied by the Generating Function Enumerating Words with exactly one pattern 123, and of length`, 3*n
`in the alphabet {1, ...,n}, where Each Letter Appears Exactly`, 3, ` Times . `
`Theorem: Let a(n) be the number of words in the alphabet {1, ...,n} where each letter occurs exactly`, 3, `times `
`and that have exactly one pattern 123, in other words, there is exactly one i1<i2<i3 such that w[i1]<w[i2]<w[i3] `
`Let F be the ordinary generating function `
``
F = Sum(a(n)*x^n, n = 0 .. infinity)
`then F satisfies the following algebraic equation (derived rigorously via the Yang scheme and Groebner Bases)`
``
16*x^3*(4*x+3)*(64*x^7-12304*x^6+1168160*x^5+58072*x^4-1483056*x^3+201042*x^2+676863*x+11664)*(64*x^6-80*x^5+128*x^4-72*x^3+1248*x^2+166*x-1)+(4194304*x^17-316735488*x^16-10919641088*x^15-11607773184*x^14-33879937024*x^13+43040624640*x^12-111269412864*x^11-336643549184*x^10+167590306816*x^9+420849985280*x^8-138443655552*x^7-347562926912*x^6-105480829952*x^5+5333115152*x^4+1652378400*x^3-40804560*x^2-830520*x+5832)*F+4*x*(7340032*x^16-106381312*x^15-948744192*x^14+5698564096*x^13+29244638208*x^12-33632006144*x^11-114402076416*x^10-42761499136*x^9+1080847424*x^8+5202950768*x^7+106577929632*x^6+134513565488*x^5+53164591120*x^4+5561083488*x^3+186295860*x^2+2536569*x-18225)*F^2+2*x^2*(58720256*x^15+581959680*x^14+8706158592*x^13+13878194176*x^12+11987657472*x^11+49028197632*x^10+137497145088*x^9+139816788992*x^8-88046980928*x^7-363777262144*x^6-374283577920*x^5-179941745712*x^4-38547786828*x^3-2419968708*x^2-37764441*x+171315)*F^3+x^3*(293601280*x^14+3937927168*x^13+6148583424*x^12-42046171136*x^11-181067272448*x^10+106616392192*x^9+949540282624*x^8+1112127624384*x^7+450056230976*x^6+125791307776*x^5+154302597760*x^4+84474468016*x^3+11518738668*x^2+301482216*x-661203)*F^4+8*x^4*(58720256*x^13+459800576*x^12-1132052480*x^11-7594454272*x^10-3972674176*x^9+17025258560*x^8+27432696352*x^7+17549449992*x^6+5305148280*x^5-2218829376*x^4-3820288471*x^3-1435607406*x^2-78629022*x+729)*F^5+4*x^5*(117440512*x^12+120586240*x^11-2920644608*x^10-7695873536*x^9+15579358720*x^8+68179659280*x^7+76221365584*x^6+27913506672*x^5-3164022160*x^4-2094940124*x^3+757631061*x^2+160784244*x+473850)*F^6+4*x^6*(67108864*x^11-222298112*x^10-1645084672*x^9-867139584*x^8+2710728640*x^7+2714202112*x^6+1254824368*x^5+2581628232*x^4+2498103550*x^3+636041502*x^2-46912365*x-531441)*F^7+2*x^7*(33554432*x^10-239075328*x^9-614858752*x^8+3253329920*x^7+11316039136*x^6+12249556904*x^5+4242483584*x^4-1084831864*x^3-895682934*x^2-97245684*x-492075)*F^8-8*x^8*(32*x+27)*(458752*x^7-22528*x^6+913728*x^5+3963198*x^4+2535675*x^3-773758*x^2-731241*x-13122)*F^9+4*x^9*(4096*x^5+45568*x^4+62508*x^3+17167*x^2-5013*x-243)*(32*x+27)^2*F^10-18*x^10*(32*x^2+31*x+3)*(32*x+27)^3*F^11+x^11*(32*x+27)^4*F^12 = 0
``
`This ends this article, that took`, 20.522, `seconds to generate `