0132_0213_0321_2103
Counting sequence:
1, 1, 2, 6, 20, 65, 203, 621, 1891, 5770, 17653, 54086, 165778, 508108, 1557197, 4772114, 14624364, 44817632, 137349145, 420925968, 1289990267, 3953368962, 12115693647, 37130367559, 113791616159, 348731624372, 1068740909259, 3275318611110, 10037710848030, 30762088195703, 94275088104383, 288920316658712, 885440164704477, 2713565786300882, 8316134252397291, 25486055758272668, 78105886527430034, 239367345377866997, 733577564825956176, 2248159801368191503, 6889826972577920573, 21114920604793773375, 64709879351855398954, 198313247969410953546, 607761051545878278460, 1862575998118566769782, 5708146877702299182448, 17493482580229388574811, 53611432806730941046414, 164300373834052130008291, 503523435744157950735493, 1543124001651361364880252, 4729137743019591234057377, 14493160477394854510751078, 44416490285902130037844075, 136121076724065045383142247, 417163702247766253597689820, 1278461489294825404886206069, 3918039299208221376871236107, 12007426174884152796581580765, 36798580191482375521041791571, 112774834871889110269760498737, 345615600227034899814647986145, 1059191469940856739477037229587, 3246053040599167274260979710119, 9948022280590550727640866724198, 30487224348268528783860220007082, 93432726852163892033229362064358, 286338774147107786323171052644631, 877528638437352055416599394251591, 2689319717776304659061601447069779, 8241828502941510056603241020232501, 25258334523373789743234143978457084, 77408000259524914804681903786232369, 237228567015520308061402496441687064, 727022953952519632381844795320241608, 2228072201520599813954596724198019292, 6828265473875341909785332944854681159, 20926256047670980571174446987097263510, 64131688178806860264177500997617578004, 196541293353880379010138750633257506622, 602330627653449808926233125518237779701, 1845933639788149249630420454423737243840, 5657143844363845012233914170101346195029, 17337176042523737328021519917355041236972, 53132407695258009555376094155001017789765, 162832328665918450507583084569398177726703, 499024388483190390671469817273189063754339, 1529335988383148772082480518239148442204492, 4686882283394947171218501020791592355765381, 14363662207168314115996988407727144036605954, 44019623179481911346714505709549887467058356, 134904817233625911372767768039275163832565473, 413436290416064559912598249039574323444068253, 1267038269930594885413876028382701897385199194, 3883031109468216729809302551080663989796432681, 11900138263324832797815620864710900582462507710, 36469780100639416846414424466888467221865254301, 111767177082981859193758980126412051639430341760, 342527479974566873092449743980005395992283264904, 1049727456663050318106219356365677522283204750709
Generating function in Maple syntax:
-(x^3-2*x^2+3*x-1)*(x-1)^4/(x^8-5*x^7+17*x^6-36*x^5+52*x^4-47*x^3+26*x^2-8*x+1)
Generating function in latex syntax:
-\frac{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{4}}{x^{8}-5 x^{7}+17 x^{6}-36 x^{5}+52 x^{4}-47 x^{3}+26 x^{2}-8 x +1}
Generating function in sympy syntax:
(x - 1)**4*(-x**3 + 2*x**2 - 3*x + 1)/(x**8 - 5*x**7 + 17*x**6 - 36*x**5 + 52*x**4 - 47*x**3 + 26*x**2 - 8*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^8-5*x^7+17*x^6-36*x^5+52*x^4-47*x^3+26*x^2-8*x+1)*F(x)+(x^3-2*x^2+3*x-1)*(x-1)^4 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{8}-5 x^{7}+17 x^{6}-36 x^{5}+52 x^{4}-47 x^{3}+26 x^{2}-8 x +1\right) F \! \left(x \right)+\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{4} = 0
Explicit closed form in Maple syntax:
-7419665/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n+6)-7419665/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n+6)-7419665/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n+6)-7419665/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n+6)-7419665/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n+6)-7419665/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n+6)-7419665/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 7)^(-n+6)-7419665/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 8)^(-n+6)+32425002/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n+5)+32425002/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n+5)+32425002/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n+5)+32425002/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n+5)+32425002/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n+5)+32425002/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n+5)+32425002/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 7)^(-n+5)+32425002/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 8)^(-n+5)-104746494/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n+4)-104746494/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n+4)-104746494/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n+4)-104746494/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n+4)-104746494/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n+4)-104746494/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n+4)-104746494/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 7)^(-n+4)-104746494/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 8)^(-n+4)+198697213/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n+3)+198697213/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n+3)+198697213/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n+3)+198697213/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n+3)+198697213/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n+3)+198697213/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n+3)+198697213/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 7)^(-n+3)+198697213/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 8)^(-n+3)-253675538/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n+2)-253675538/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n+2)-253675538/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n+2)-253675538/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n+2)-253675538/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n+2)-253675538/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n+2)-253675538/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 7)^(-n+2)-253675538/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 8)^(-n+2)+179784201/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n+1)+179784201/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n+1)+179784201/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n+1)+179784201/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n+1)+179784201/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n+1)+179784201/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n+1)+179784201/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 7)^(-n+1)+179784201/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 8)^(-n+1)+12878138/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n-1)+12878138/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n-1)+12878138/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n-1)+12878138/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n-1)+12878138/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n-1)+12878138/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n-1)+12878138/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 7)^(-n-1)+12878138/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 8)^(-n-1)-69021328/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 1)^(-n)-69021328/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 2)^(-n)-69021328/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 3)^(-n)-69021328/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 4)^(-n)-69021328/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 5)^(-n)-69021328/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 6)^(-n)-69021328/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 7)^(-n)-69021328/32819431*RootOf(_Z^8-5*_Z^7+17*_Z^6-36*_Z^5+52*_Z^4-47*_Z^3+26*_Z^2-8*_Z+1,index = 8)^(-n)
Explicit closed form in latex syntax:
-\frac{7419665 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +6}}{32819431}-\frac{7419665 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +6}}{32819431}-\frac{7419665 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +6}}{32819431}-\frac{7419665 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +6}}{32819431}-\frac{7419665 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +6}}{32819431}-\frac{7419665 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +6}}{32819431}-\frac{7419665 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +6}}{32819431}-\frac{7419665 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +6}}{32819431}+\frac{32425002 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{32819431}+\frac{32425002 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{32819431}+\frac{32425002 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{32819431}+\frac{32425002 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{32819431}+\frac{32425002 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{32819431}+\frac{32425002 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{32819431}+\frac{32425002 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{32819431}+\frac{32425002 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +5}}{32819431}-\frac{104746494 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{32819431}-\frac{104746494 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{32819431}-\frac{104746494 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{32819431}-\frac{104746494 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{32819431}-\frac{104746494 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{32819431}-\frac{104746494 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{32819431}-\frac{104746494 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{32819431}-\frac{104746494 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +4}}{32819431}+\frac{198697213 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{32819431}+\frac{198697213 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{32819431}+\frac{198697213 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{32819431}+\frac{198697213 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{32819431}+\frac{198697213 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{32819431}+\frac{198697213 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{32819431}+\frac{198697213 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{32819431}+\frac{198697213 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +3}}{32819431}-\frac{253675538 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{32819431}-\frac{253675538 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{32819431}-\frac{253675538 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{32819431}-\frac{253675538 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{32819431}-\frac{253675538 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{32819431}-\frac{253675538 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{32819431}-\frac{253675538 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{32819431}-\frac{253675538 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +2}}{32819431}+\frac{179784201 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{32819431}+\frac{179784201 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{32819431}+\frac{179784201 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{32819431}+\frac{179784201 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{32819431}+\frac{179784201 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{32819431}+\frac{179784201 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{32819431}+\frac{179784201 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{32819431}+\frac{179784201 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +1}}{32819431}+\frac{12878138 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{32819431}+\frac{12878138 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{32819431}+\frac{12878138 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{32819431}+\frac{12878138 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{32819431}+\frac{12878138 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{32819431}+\frac{12878138 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{32819431}+\frac{12878138 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{32819431}+\frac{12878138 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =8\right)^{-n -1}}{32819431}-\frac{69021328 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{32819431}-\frac{69021328 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{32819431}-\frac{69021328 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{32819431}-\frac{69021328 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{32819431}-\frac{69021328 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{32819431}-\frac{69021328 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{32819431}-\frac{69021328 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{32819431}-\frac{69021328 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-5 \textit{\_Z}^{7}+17 \textit{\_Z}^{6}-36 \textit{\_Z}^{5}+52 \textit{\_Z}^{4}-47 \textit{\_Z}^{3}+26 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =8\right)^{-n}}{32819431}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(6) = 203
a(7) = 621
a(n+8) = -a(n)+5*a(n+1)-17*a(n+2)+36*a(n+3)-52*a(n+4)+47*a(n+5)-26*a(n+6)+8*a(n+7), n >= 8
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 20
a \! \left(5\right) = 65
a \! \left(6\right) = 203
a \! \left(7\right) = 621
a \! \left(n +8\right) = -a \! \left(n \right)+5 a \! \left(n +1\right)-17 a \! \left(n +2\right)+36 a \! \left(n +3\right)-52 a \! \left(n +4\right)+47 a \! \left(n +5\right)-26 a \! \left(n +6\right)+8 a \! \left(n +7\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15254/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[16,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[11,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[4,x]
F[16,x] = F[17,x]+F[2,x]
F[17,x] = F[18,x]+F[19,x]+F[72,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[36,x]
F[21,x] = F[22,x]+F[7,x]
F[22,x] = F[18,x]+F[23,x]+F[31,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[11,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[25,x]*F[4,x]
F[28,x] = F[14,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[28,x]*F[4,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[13,x]+F[33,x]
F[33,x] = F[26,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[33,x]*F[4,x]
F[36,x] = F[37,x]+F[46,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[40,x]
F[43,x] = F[41,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[43,x]
F[46,x] = 2*F[18,x]+F[47,x]+F[67,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[41,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[49,x]
F[52,x] = F[53,x]+F[65,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[53,x]+F[56,x]
F[56,x] = F[18,x]+F[57,x]+F[63,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[61,x]
F[59,x] = F[11,x]+F[60,x]
F[60,x] = F[15,x]+F[18,x]+F[23,x]
F[61,x] = F[41,x]+F[62,x]
F[62,x] = 2*F[18,x]+F[47,x]+F[54,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[2,x]+F[56,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[52,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[55,x]+F[69,x]
F[69,x] = F[50,x]+F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[69,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[64,x]+F[74,x]
F[74,x] = F[56,x]+F[75,x]
F[75,x] = 2*F[18,x]+F[76,x]+F[82,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[78,x]+F[80,x]
F[78,x] = F[14,x]+F[79,x]
F[79,x] = F[35,x]
F[80,x] = F[44,x]+F[81,x]
F[81,x] = F[71,x]
F[82,x] = F[4,x]*F[74,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{16}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{2}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{72}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{36}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{31}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{26}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{46}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{47}\! \left(x \right)+F_{67}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{41}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{65}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{53}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{18}\! \left(x \right)+F_{57}\! \left(x \right)+F_{63}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{61}\! \left(x \right)
F_{59}\! \left(x \right) = F_{11}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{23}\! \left(x \right)
F_{61}\! \left(x \right) = F_{41}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{47}\! \left(x \right)+F_{54}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{2}\! \left(x \right)+F_{56}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{55}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{50}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{64}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{56}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{76}\! \left(x \right)+F_{82}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{80}\! \left(x \right)
F_{78}\! \left(x \right) = F_{14}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{35}\! \left(x \right)
F_{80}\! \left(x \right) = F_{44}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{71}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_16(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_11(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_2(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_72(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_36(x))
Eq(F_21(x), F_22(x) + F_7(x))
Eq(F_22(x), F_18(x) + F_23(x) + F_31(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_11(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_25(x)*F_4(x))
Eq(F_28(x), F_14(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_28(x)*F_4(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_13(x) + F_33(x))
Eq(F_33(x), F_26(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_33(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_46(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_2(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_40(x))
Eq(F_43(x), F_41(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_43(x))
Eq(F_46(x), 2*F_18(x) + F_47(x) + F_67(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_41(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_49(x))
Eq(F_52(x), F_53(x) + F_65(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_53(x) + F_56(x))
Eq(F_56(x), F_18(x) + F_57(x) + F_63(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_61(x))
Eq(F_59(x), F_11(x) + F_60(x))
Eq(F_60(x), F_15(x) + F_18(x) + F_23(x))
Eq(F_61(x), F_41(x) + F_62(x))
Eq(F_62(x), 2*F_18(x) + F_47(x) + F_54(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_2(x) + F_56(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_52(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_55(x) + F_69(x))
Eq(F_69(x), F_50(x) + F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_69(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_64(x) + F_74(x))
Eq(F_74(x), F_56(x) + F_75(x))
Eq(F_75(x), 2*F_18(x) + F_76(x) + F_82(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_78(x) + F_80(x))
Eq(F_78(x), F_14(x) + F_79(x))
Eq(F_79(x), F_35(x))
Eq(F_80(x), F_44(x) + F_81(x))
Eq(F_81(x), F_71(x))
Eq(F_82(x), F_4(x)*F_74(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 2], [1, 0], [1, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": 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