0123_0231_1302_2013
Counting sequence:
1, 1, 2, 6, 20, 65, 204, 629, 1929, 5911, 18111, 55476, 169880, 520136, 1592538, 4876282, 14931918, 45725786, 140028449, 428819819, 1313209243, 4021540621, 12315449490, 37714426107, 115495316912, 353688590488, 1083122658132, 3316913980823, 10157592398843, 31106229180364, 95258549782171, 291716216458597, 893340829629262, 2735733555112620, 8377808164851302, 25655886521847023, 78567627699833010, 240602565551350657, 736812300678932317, 2256386439796044125, 6909873464871595473, 21160538131359767786, 64801240758315533694, 198444896712602188140, 607710231639622338642, 1861029090498378277836, 5699145901094731934124, 17452851311081696549660, 53446959276579941788575, 163673969656710371525123, 501229044753535341822147, 1534945085228932774742551, 4700558435967075955277848, 14394814396009941534407433, 44082141370704754331284434, 134995501460955097983187208, 413405176065366598332442736, 1265996553574548816770395749, 3876940508865861860308099353, 11872597651901956499246858834, 36358199121601564406644242669, 111341989522761560460560882289, 340969545532888608481487918914, 1044172387068201565702363567078, 3197634475570899316739545111936, 9792316255430462757653796091281, 29987623156723614222419946055112, 91832975889741175531609471040589, 281225871643479333969845415072305, 861215594021382157327038607140199, 2637354433470735169339506444734171, 8076535603900187507309234921687036, 24733280644128176447870018597430844, 75742273843972957154284018014715448, 231950307344988674191454522930205570, 710315948373337772556517748701823362, 2175245000917715375821057193320932978, 6661388956355747558164819167900657442, 20399588464351049952417315520843103389, 62470936953446461552957454285593256731, 191308661479197758515550567739517311819, 585856491703269586158525634369671560165, 1794105014467338119941235650374884443238, 5494200113032363218886957407315154389715, 16825232992845178022632572632674581236924, 51524964406016784023926464263502004318608, 157788124430148602741383104115670742707956, 483204452408640994443354906968771706657607, 1479747247587678961037588811089711244252287, 4531522641872400919144621200879978295965276, 13877165500580185037847588116867480229203295, 42496912748718306986219216388138823757316157, 130140956602172881364865861733265522115193526, 398538799405833301943271997791768214994222564, 1220470317560206849260639326618877318456343234, 3737522665964326817031160279559096443556890623, 11445649662764520606406432129549265811035291126, 35050729563653689792530279465774692125319789313, 107338043636017472315192719437032746963576941597, 328708011360622699772583816209594412686315028581, 1006623123289338966324652524215656386420489044445
Generating function in Maple syntax:
-(2*x-1)*(x-1)^6/(x^8-9*x^7+34*x^6-74*x^5+93*x^4-72*x^3+34*x^2-9*x+1)
Generating function in latex syntax:
-\frac{\left(2 x -1\right) \left(x -1\right)^{6}}{x^{8}-9 x^{7}+34 x^{6}-74 x^{5}+93 x^{4}-72 x^{3}+34 x^{2}-9 x +1}
Generating function in sympy syntax:
(1 - 2*x)*(x - 1)**6/(x**8 - 9*x**7 + 34*x**6 - 74*x**5 + 93*x**4 - 72*x**3 + 34*x**2 - 9*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^8-9*x^7+34*x^6-74*x^5+93*x^4-72*x^3+34*x^2-9*x+1)*F(x)+(2*x-1)*(x-1)^6 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{8}-9 x^{7}+34 x^{6}-74 x^{5}+93 x^{4}-72 x^{3}+34 x^{2}-9 x +1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right)^{6} = 0
Explicit closed form in Maple syntax:
-515069/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 1)^(-n+6)-515069/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 2)^(-n+6)-515069/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 3)^(-n+6)-515069/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 4)^(-n+6)-515069/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 5)^(-n+6)-515069/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 6)^(-n+6)-515069/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 7)^(-n+6)-515069/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 8)^(-n+6)+4704015/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 1)^(-n+5)+4704015/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 2)^(-n+5)+4704015/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 3)^(-n+5)+4704015/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 4)^(-n+5)+4704015/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 5)^(-n+5)+4704015/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 6)^(-n+5)+4704015/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 7)^(-n+5)+4704015/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 8)^(-n+5)-17903387/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 1)^(-n+4)-17903387/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 2)^(-n+4)-17903387/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 3)^(-n+4)-17903387/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 4)^(-n+4)-17903387/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 5)^(-n+4)-17903387/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 6)^(-n+4)-17903387/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 7)^(-n+4)-17903387/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 8)^(-n+4)+19347887/1969234*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 1)^(-n+3)+19347887/1969234*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 2)^(-n+3)+19347887/1969234*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 3)^(-n+3)+19347887/1969234*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 4)^(-n+3)+19347887/1969234*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 5)^(-n+3)+19347887/1969234*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 6)^(-n+3)+19347887/1969234*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 7)^(-n+3)+19347887/1969234*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 8)^(-n+3)-23667039/1969234*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 1)^(-n+2)-23667039/1969234*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 2)^(-n+2)-23667039/1969234*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 3)^(-n+2)-23667039/1969234*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 4)^(-n+2)-23667039/1969234*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 5)^(-n+2)-23667039/1969234*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 6)^(-n+2)-23667039/1969234*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 7)^(-n+2)-23667039/1969234*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 8)^(-n+2)+32902077/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 1)^(-n+1)+32902077/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 2)^(-n+1)+32902077/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 3)^(-n+1)+32902077/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 4)^(-n+1)+32902077/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 5)^(-n+1)+32902077/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 6)^(-n+1)+32902077/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 7)^(-n+1)+32902077/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 8)^(-n+1)+2123765/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 1)^(-n-1)+2123765/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 2)^(-n-1)+2123765/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 3)^(-n-1)+2123765/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 4)^(-n-1)+2123765/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 5)^(-n-1)+2123765/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 6)^(-n-1)+2123765/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 7)^(-n-1)+2123765/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 8)^(-n-1)-12378149/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 1)^(-n)-12378149/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 2)^(-n)-12378149/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 3)^(-n)-12378149/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 4)^(-n)-12378149/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 5)^(-n)-12378149/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 6)^(-n)-12378149/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 7)^(-n)-12378149/3938468*RootOf(_Z^8-9*_Z^7+34*_Z^6-74*_Z^5+93*_Z^4-72*_Z^3+34*_Z^2-9*_Z+1,index = 8)^(-n)
Explicit closed form in latex syntax:
-\frac{515069 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +6}}{3938468}-\frac{515069 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +6}}{3938468}-\frac{515069 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +6}}{3938468}-\frac{515069 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +6}}{3938468}-\frac{515069 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +6}}{3938468}-\frac{515069 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +6}}{3938468}-\frac{515069 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +6}}{3938468}-\frac{515069 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +6}}{3938468}+\frac{4704015 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +5}}{3938468}+\frac{4704015 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +5}}{3938468}+\frac{4704015 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +5}}{3938468}+\frac{4704015 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +5}}{3938468}+\frac{4704015 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +5}}{3938468}+\frac{4704015 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +5}}{3938468}+\frac{4704015 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +5}}{3938468}+\frac{4704015 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +5}}{3938468}-\frac{17903387 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{3938468}-\frac{17903387 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{3938468}-\frac{17903387 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{3938468}-\frac{17903387 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{3938468}-\frac{17903387 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{3938468}-\frac{17903387 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{3938468}-\frac{17903387 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +4}}{3938468}-\frac{17903387 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +4}}{3938468}+\frac{19347887 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{1969234}+\frac{19347887 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{1969234}+\frac{19347887 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{1969234}+\frac{19347887 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{1969234}+\frac{19347887 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{1969234}+\frac{19347887 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{1969234}+\frac{19347887 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +3}}{1969234}+\frac{19347887 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +3}}{1969234}-\frac{23667039 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{1969234}-\frac{23667039 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{1969234}-\frac{23667039 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{1969234}-\frac{23667039 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{1969234}-\frac{23667039 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{1969234}-\frac{23667039 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{1969234}-\frac{23667039 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +2}}{1969234}-\frac{23667039 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +2}}{1969234}+\frac{32902077 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{3938468}+\frac{32902077 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{3938468}+\frac{32902077 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{3938468}+\frac{32902077 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{3938468}+\frac{32902077 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{3938468}+\frac{32902077 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{3938468}+\frac{32902077 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =7\right)^{-n +1}}{3938468}+\frac{32902077 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =8\right)^{-n +1}}{3938468}+\frac{2123765 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{3938468}+\frac{2123765 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{3938468}+\frac{2123765 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{3938468}+\frac{2123765 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{3938468}+\frac{2123765 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{3938468}+\frac{2123765 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{3938468}+\frac{2123765 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =7\right)^{-n -1}}{3938468}+\frac{2123765 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =8\right)^{-n -1}}{3938468}-\frac{12378149 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{3938468}-\frac{12378149 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{3938468}-\frac{12378149 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{3938468}-\frac{12378149 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{3938468}-\frac{12378149 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{3938468}-\frac{12378149 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{3938468}-\frac{12378149 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =7\right)^{-n}}{3938468}-\frac{12378149 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-9 \textit{\_Z}^{7}+34 \textit{\_Z}^{6}-74 \textit{\_Z}^{5}+93 \textit{\_Z}^{4}-72 \textit{\_Z}^{3}+34 \textit{\_Z}^{2}-9 \textit{\_Z} +1, \mathit{index} =8\right)^{-n}}{3938468}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(6) = 204
a(7) = 629
a(n+8) = -a(n)+9*a(n+1)-34*a(n+2)+74*a(n+3)-93*a(n+4)+72*a(n+5)-34*a(n+6)+9*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) = 204
a \! \left(7\right) = 629
a \! \left(n +8\right) = -a \! \left(n \right)+9 a \! \left(n +1\right)-34 a \! \left(n +2\right)+74 a \! \left(n +3\right)-93 a \! \left(n +4\right)+72 a \! \left(n +5\right)-34 a \! \left(n +6\right)+9 a \! \left(n +7\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15013/
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[12,x]*F[4,x]
F[4,x] = F[21,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[10,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[10,x]+F[14,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[10,x]+F[20,x]
F[20,x] = F[18,x]
F[21,x] = F[2,x]+F[22,x]
F[22,x] = F[15,x]+F[23,x]+F[50,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[34,x]
F[25,x] = F[26,x]+F[6,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[10,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[10,x]+F[31,x]
F[34,x] = F[22,x]+F[35,x]
F[35,x] = 2*F[15,x]+F[36,x]+F[40,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[26,x]
F[39,x] = F[35,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[46,x]
F[42,x] = F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[2,x]+F[43,x]
F[46,x] = F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[43,x]+F[47,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[45,x]+F[52,x]
F[52,x] = F[53,x]+F[66,x]
F[53,x] = F[15,x]+F[54,x]+F[64,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[58,x]
F[56,x] = F[10,x]+F[57,x]
F[57,x] = F[27,x]
F[58,x] = F[53,x]+F[59,x]
F[59,x] = 2*F[15,x]+F[40,x]+F[60,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[57,x]
F[63,x] = F[59,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[2,x]+F[53,x]
F[66,x] = 2*F[15,x]+F[67,x]+F[69,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[43,x]+F[66,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[53,x]+F[71,x]
F[71,x] = F[69,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{21}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{2}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{50}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{34}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{6}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{22}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{26}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{46}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{2}\! \left(x \right)+F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{43}\! \left(x \right)+F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{45}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{66}\! \left(x \right)
F_{53}\! \left(x \right) = F_{15}\! \left(x \right)+F_{54}\! \left(x \right)+F_{64}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{58}\! \left(x \right)
F_{56}\! \left(x \right) = F_{10}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{27}\! \left(x \right)
F_{58}\! \left(x \right) = F_{53}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{57}\! \left(x \right)
F_{63}\! \left(x \right) = F_{59}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{2}\! \left(x \right)+F_{53}\! \left(x \right)
F_{66}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{67}\! \left(x \right)+F_{69}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{43}\! \left(x \right)+F_{66}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{53}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{69}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_21(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_10(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_10(x) + F_14(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_10(x) + F_20(x))
Eq(F_20(x), F_18(x))
Eq(F_21(x), F_2(x) + F_22(x))
Eq(F_22(x), F_15(x) + F_23(x) + F_50(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_34(x))
Eq(F_25(x), F_26(x) + F_6(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_10(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_10(x) + F_31(x))
Eq(F_34(x), F_22(x) + F_35(x))
Eq(F_35(x), 2*F_15(x) + F_36(x) + F_40(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_26(x))
Eq(F_39(x), F_35(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_46(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_2(x) + F_43(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_43(x) + F_47(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_45(x) + F_52(x))
Eq(F_52(x), F_53(x) + F_66(x))
Eq(F_53(x), F_15(x) + F_54(x) + F_64(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_58(x))
Eq(F_56(x), F_10(x) + F_57(x))
Eq(F_57(x), F_27(x))
Eq(F_58(x), F_53(x) + F_59(x))
Eq(F_59(x), 2*F_15(x) + F_40(x) + F_60(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_57(x))
Eq(F_63(x), F_59(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_2(x) + F_53(x))
Eq(F_66(x), 2*F_15(x) + F_67(x) + F_69(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_43(x) + F_66(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_53(x) + F_71(x))
Eq(F_71(x), F_69(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}], "requirements": []}, {"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]]}]]}], "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": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 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, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], 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