0231_0312_0321_2130
Counting sequence:
1, 1, 2, 6, 20, 65, 209, 673, 2169, 6990, 22525, 72586, 233907, 753761, 2428980, 7827340, 25223449, 81282068, 261929864, 844063830, 2719979075, 8765079021, 28245294587, 91019905741, 293309854340, 945185231215, 3045840799714, 9815162013563, 31629166357459, 101924365903020, 328449262535900, 1058421282336181, 3410741745172042, 10991048126490946, 35418436206681100, 114135213392723565, 367798478176324629, 1185223355068827033, 3819358928198609329, 12307808954341200810, 39661672051337581125, 127808957365478789066, 411861848933317378147, 1327216699856968497501, 4276929686353200651760, 13782321714291715500640, 44413260391522400875069, 143120857247130572426228, 461204144856328348931724, 1486221277066321564045950, 4789318806084748237865375, 15433485565214031983885801, 49734103394631067428681027, 160267169073129020325819781, 516457797155899873183096320, 1664275083822175786390536815, 5363093693007416467251931034, 17282463842407178168447805523, 55692399492021785951307586519, 179467660945902366574817158440, 578331004215528872528599017400, 1863660275473037367656860030761, 6005608547802938622071521995682, 19352955312785775977846581180286, 62364517493518931541382062619780, 200968429851635709235946091881465, 647616808728281618285602960000649, 2086932416484668493917419586421793, 6725098626650795831765533005648089, 21671497927260586441103751028827030, 69835975423479377929431695219325325, 225045055940224393945743571255693946, 725203262301845150551203748830383987, 2336953235679754793953578387958629641, 7530785794343192547681654001606602140, 24267808963573492338093113073736268340, 78202536624381831752737699745956448289, 252006134697511635147529985629041178788, 812084807814025082666457837036202675184, 2616927305654412403842942023674257190470, 8432996722982642313576623742026810377275, 27175165919273488741210316885781877660981, 87571437176943316113983292541730350039067, 282197232282449645372946067132918928062221, 909375025408878555914835458912347349885900, 2930443116499795788713683278706284627396815, 9443295251241229973000925262926422088353954, 30430833036823896676205392773736879892895883, 98062760368880160161958380337535556283367179, 316005314725622736220590674817165797654206260, 1018320905501757231987183735261897574836764500, 3281519070279855912961673359234500716969923741, 10574630600659693371488159807675369302501550922, 34076538927708217314659437568367068854832388026, 109810975829186121001888843158832105573273776060, 353863707759157605661604522451081556041794353765, 1140318831733548944985620602297654175137549388269, 3674654985786727294275298930778993805414028104953, 11841503348707683215150994680397976633898946798449, 38159011417349305141085829768240319808013733525650, 122966663055354918956793707768576601210006806175125
Generating function in Maple syntax:
(x^4+2*x^3+x^2+2*x-1)/(x^4+2*x^3+3*x-1)
Generating function in latex syntax:
\frac{x^{4}+2 x^{3}+x^{2}+2 x -1}{x^{4}+2 x^{3}+3 x -1}
Generating function in sympy syntax:
(x**4 + 2*x**3 + x**2 + 2*x - 1)/(x**4 + 2*x**3 + 3*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^4+2*x^3+3*x-1)*F(x)-x^4-2*x^3-x^2-2*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{4}+2 x^{3}+3 x -1\right) F \! \left(x \right)-x^{4}-2 x^{3}-x^{2}-2 x +1 = 0
Explicit closed form in Maple syntax:
46/935*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^4+2*_Z^3+3*_Z-1))+35/187*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^4+2*_Z^3+3*_Z-1))+19/935*Sum(_alpha^(-n),_alpha = RootOf(_Z^4+2*_Z^3+3*_Z-1))+6/187*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^4+2*_Z^3+3*_Z-1))+piecewise(n = 0,1,0)
Explicit closed form in latex syntax:
\frac{46 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{935}+\frac{35 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{187}+\frac{19 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{935}+\frac{6 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+2 \textit{\_Z}^{3}+3 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{187}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+1) = -1/2*a(n)-3/2*a(n+3)+1/2*a(n+4), n >= 5
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(n +1\right) = -\frac{a \! \left(n \right)}{2}-\frac{3 a \! \left(n +3\right)}{2}+\frac{a \! \left(n +4\right)}{2}, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15772/
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[10,x]*F[4,x]
F[4,x] = F[13,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = x
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[6,x]
F[13,x] = F[14,x]+F[29,x]
F[14,x] = F[15,x]
F[15,x] = F[10,x]*F[16,x]
F[16,x] = F[17,x]+F[22,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[10,x]*F[20,x]
F[20,x] = F[17,x]+F[21,x]
F[21,x] = F[10,x]
F[22,x] = F[14,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[10,x]*F[25,x]
F[25,x] = F[22,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[10,x]*F[14,x]
F[29,x] = F[30,x]+F[31,x]+F[50,x]
F[30,x] = 0
F[31,x] = F[10,x]*F[32,x]
F[32,x] = F[33,x]+F[40,x]
F[33,x] = F[34,x]+F[6,x]
F[34,x] = F[30,x]+F[35,x]+F[38,x]
F[35,x] = F[10,x]*F[36,x]
F[36,x] = F[33,x]+F[37,x]
F[37,x] = F[11,x]
F[38,x] = F[10,x]*F[39,x]
F[39,x] = F[9,x]
F[40,x] = F[29,x]+F[41,x]
F[41,x] = 2*F[30,x]+F[42,x]+F[47,x]
F[42,x] = F[10,x]*F[43,x]
F[43,x] = F[40,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[10,x]*F[29,x]
F[47,x] = F[10,x]*F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[27,x]+F[45,x]
F[50,x] = F[10,x]*F[51,x]
F[51,x] = F[13,x]+F[52,x]
F[52,x] = F[53,x]+F[66,x]
F[53,x] = F[28,x]+F[30,x]+F[54,x]
F[54,x] = F[10,x]*F[55,x]
F[55,x] = F[56,x]+F[61,x]
F[56,x] = F[10,x]+F[57,x]
F[57,x] = F[30,x]+F[58,x]+F[60,x]
F[58,x] = F[10,x]*F[59,x]
F[59,x] = F[56,x]
F[60,x] = x^2
F[61,x] = F[27,x]+F[62,x]
F[62,x] = 2*F[30,x]+F[63,x]+F[65,x]
F[63,x] = F[10,x]*F[64,x]
F[64,x] = F[61,x]
F[65,x] = F[10,x]*F[27,x]
F[66,x] = 2*F[30,x]+F[46,x]+F[67,x]
F[67,x] = F[10,x]*F[68,x]
F[68,x] = F[69,x]+F[74,x]
F[69,x] = F[11,x]+F[70,x]
F[70,x] = 2*F[30,x]+F[71,x]+F[73,x]
F[71,x] = F[10,x]*F[72,x]
F[72,x] = F[69,x]
F[73,x] = F[10,x]*F[11,x]
F[74,x] = F[45,x]+F[75,x]
F[75,x] = 3*F[30,x]+F[76,x]+F[78,x]
F[76,x] = F[10,x]*F[77,x]
F[77,x] = F[74,x]
F[78,x] = F[10,x]*F[45,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{13}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{11}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{6}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{29}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{22}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \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_{17}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right)
F_{22}\! \left(x \right) = F_{14}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{50}\! \left(x \right)
F_{30}\! \left(x \right) = 0
F_{31}\! \left(x \right) = F_{10}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{6}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{10}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{11}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{9}\! \left(x \right)
F_{40}\! \left(x \right) = F_{29}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{30}\! \left(x \right)+F_{42}\! \left(x \right)+F_{47}\! \left(x \right)
F_{42}\! \left(x \right) = F_{10}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \left(x \right) F_{29}\! \left(x \right)
F_{47}\! \left(x \right) = F_{10}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{27}\! \left(x \right)+F_{45}\! \left(x \right)
F_{50}\! \left(x \right) = F_{10}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{13}\! \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_{28}\! \left(x \right)+F_{30}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{10}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{61}\! \left(x \right)
F_{56}\! \left(x \right) = F_{10}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{30}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)
F_{58}\! \left(x \right) = F_{10}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{56}\! \left(x \right)
F_{60}\! \left(x \right) = x^{2}
F_{61}\! \left(x \right) = F_{27}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = 2 F_{30}\! \left(x \right)+F_{63}\! \left(x \right)+F_{65}\! \left(x \right)
F_{63}\! \left(x \right) = F_{10}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{61}\! \left(x \right)
F_{65}\! \left(x \right) = F_{10}\! \left(x \right) F_{27}\! \left(x \right)
F_{66}\! \left(x \right) = 2 F_{30}\! \left(x \right)+F_{46}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{10}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{74}\! \left(x \right)
F_{69}\! \left(x \right) = F_{11}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = 2 F_{30}\! \left(x \right)+F_{71}\! \left(x \right)+F_{73}\! \left(x \right)
F_{71}\! \left(x \right) = F_{10}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{69}\! \left(x \right)
F_{73}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right)
F_{74}\! \left(x \right) = F_{45}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = 3 F_{30}\! \left(x \right)+F_{76}\! \left(x \right)+F_{78}\! \left(x \right)
F_{76}\! \left(x \right) = F_{10}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{74}\! \left(x \right)
F_{78}\! \left(x \right) = F_{10}\! \left(x \right) F_{45}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_13(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_10(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_6(x))
Eq(F_13(x), F_14(x) + F_29(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_10(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_22(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_10(x)*F_20(x))
Eq(F_20(x), F_17(x) + F_21(x))
Eq(F_21(x), F_10(x))
Eq(F_22(x), F_14(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_10(x)*F_25(x))
Eq(F_25(x), F_22(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_10(x)*F_14(x))
Eq(F_29(x), F_30(x) + F_31(x) + F_50(x))
Eq(F_30(x), 0)
Eq(F_31(x), F_10(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_40(x))
Eq(F_33(x), F_34(x) + F_6(x))
Eq(F_34(x), F_30(x) + F_35(x) + F_38(x))
Eq(F_35(x), F_10(x)*F_36(x))
Eq(F_36(x), F_33(x) + F_37(x))
Eq(F_37(x), F_11(x))
Eq(F_38(x), F_10(x)*F_39(x))
Eq(F_39(x), F_9(x))
Eq(F_40(x), F_29(x) + F_41(x))
Eq(F_41(x), 2*F_30(x) + F_42(x) + F_47(x))
Eq(F_42(x), F_10(x)*F_43(x))
Eq(F_43(x), F_40(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_10(x)*F_29(x))
Eq(F_47(x), F_10(x)*F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_27(x) + F_45(x))
Eq(F_50(x), F_10(x)*F_51(x))
Eq(F_51(x), F_13(x) + F_52(x))
Eq(F_52(x), F_53(x) + F_66(x))
Eq(F_53(x), F_28(x) + F_30(x) + F_54(x))
Eq(F_54(x), F_10(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_61(x))
Eq(F_56(x), F_10(x) + F_57(x))
Eq(F_57(x), F_30(x) + F_58(x) + F_60(x))
Eq(F_58(x), F_10(x)*F_59(x))
Eq(F_59(x), F_56(x))
Eq(F_60(x), x**2)
Eq(F_61(x), F_27(x) + F_62(x))
Eq(F_62(x), 2*F_30(x) + F_63(x) + F_65(x))
Eq(F_63(x), F_10(x)*F_64(x))
Eq(F_64(x), F_61(x))
Eq(F_65(x), F_10(x)*F_27(x))
Eq(F_66(x), 2*F_30(x) + F_46(x) + F_67(x))
Eq(F_67(x), F_10(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_74(x))
Eq(F_69(x), F_11(x) + F_70(x))
Eq(F_70(x), 2*F_30(x) + F_71(x) + F_73(x))
Eq(F_71(x), F_10(x)*F_72(x))
Eq(F_72(x), F_69(x))
Eq(F_73(x), F_10(x)*F_11(x))
Eq(F_74(x), F_45(x) + F_75(x))
Eq(F_75(x), 3*F_30(x) + F_76(x) + F_78(x))
Eq(F_76(x), F_10(x)*F_77(x))
Eq(F_77(x), F_74(x))
Eq(F_78(x), F_10(x)*F_45(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:
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