0123_0321_1230_2031_3102

Counting sequence:
1, 1, 2, 6, 19, 46, 88, 168, 317, 589, 1090, 2012, 3707, 6825, 12560, 23108, 42509, 78193, 143826, 264544, 486579, 894965, 1646104, 3027664, 5568749, 10242533, 18838962, 34650260, 63731771, 117221009, 215603056, 396555852, 729379933, 1341538857, 2467474658, 4538393464, 8347406995, 15353275133, 28239075608, 51939757752, 95532108509, 175710941885, 323182808162, 594425858572, 1093319608635, 2010928275385, 3698673742608, 6802921626644, 12512523644653, 23014119013921, 42329564285234, 77856206943824, 143199890242995, 263385661472069, 484441758658904, 891027310373984, 1638854730504973, 3014323799537877, 5544205840416850, 10197384370459716, 18755914010414459, 34497504221291041, 63450802602165232, 116704220833870748, 214652527657327037, 394807551093363033, 726164299584560834, 1335624378335250920, 2456596229013174803, 4518384906932986573, 8310605514281412312, 15285586650227573704, 28114577071441972605, 51710769235950958637, 95110932957620504962, 174936279265013436220, 321757981458584899835, 591805193681218841033, 1088499454404817177104, 2002062629544620917988, 3682367277630656936141, 6772929361580095031249, 12457359268755372885394, 22912655907966124852800, 42142944538301592769459, 77512959715023090507669, 142568560161290808129944, 262224464414615491407088, 482305984290929390044717, 887099008866835689581765, 1631629457572380571033586, 3001034450730145650660084, 5519762917169361911275451, 10152426825471888132969137, 18673224193371395694904688, 34345413936012645739149292, 63171064954855929567023133, 116189703084239971001077129, 213706181975108546307249570, 393066950014204446875349848, 722962835073552964183676563

Generating function in Maple syntax:
(x^9-2*x^7-2*x^6+9*x^5+8*x^4+2*x^3-x+1)/(x-1)/(x^3+x^2+x-1)

Generating function in latex syntax:
\frac{x^{9}-2 x^{7}-2 x^{6}+9 x^{5}+8 x^{4}+2 x^{3}-x +1}{\left(x -1\right) \left(x^{3}+x^{2}+x -1\right)}

Generating function in sympy syntax:
(x**9 - 2*x**7 - 2*x**6 + 9*x**5 + 8*x**4 + 2*x**3 - x + 1)/((x - 1)*(x**3 + x**2 + x - 1))

Implicit equation for the generating function in Maple syntax:
-(x-1)*(x^3+x^2+x-1)*F(x)+x^9-2*x^7-2*x^6+9*x^5+8*x^4+2*x^3-x+1 = 0

Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+x^{9}-2 x^{7}-2 x^{6}+9 x^{5}+8 x^{4}+2 x^{3}-x +1 = 0

Explicit closed form in Maple syntax:
-49/11*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^4-2*_Z+1))-62/11*Sum(_alpha^(1-n),_alpha = RootOf(_Z^4-2*_Z+1))-79/22*Sum(_alpha^(-n),_alpha = RootOf(_Z^4-2*_Z+1))+125/22*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^4-2*_Z+1))+piecewise(n = 0,4,n = 1,8,n = 3,-2,n = 5,1,0)

Explicit closed form in latex syntax:
-\frac{49 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{11}-\frac{62 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{11}-\frac{79 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{22}+\frac{125 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{22}+\left(\left\{\begin{array}{cc}4 & n =0 \\ 8 & n =1 \\ -2 & n =3 \\ 1 & n =5 \\ 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) = 19
a(5) = 46
a(6) = 88
a(7) = 168
a(8) = 317
a(9) = 589
a(n+3) = a(n)+a(n+1)+a(n+2)+16, n >= 10

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) = 19
a \! \left(5\right) = 46
a \! \left(6\right) = 88
a \! \left(7\right) = 168
a \! \left(8\right) = 317
a \! \left(9\right) = 589
a \! \left(n +3\right) = a \! \left(n \right)+a \! \left(n +1\right)+a \! \left(n +2\right)+16, \quad n \geq 10

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17777/
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[26,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[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[11,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[15,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[4,x]
F[26,x] = F[27,x]+F[46,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[38,x]
F[30,x] = F[1,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[10,x]+F[34,x]
F[34,x] = F[15,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[15,x]
F[38,x] = F[27,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[27,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[27,x]
F[46,x] = F[18,x]+F[47,x]+F[70,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[55,x]
F[49,x] = F[50,x]+F[7,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[11,x]
F[54,x] = F[19,x]
F[55,x] = F[46,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[67,x]
F[59,x] = F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[65,x]
F[63,x] = F[11,x]+F[64,x]
F[64,x] = F[51,x]
F[65,x] = F[60,x]+F[66,x]
F[66,x] = F[57,x]
F[67,x] = F[68,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[60,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[72,x]+F[73,x]
F[72,x] = F[27,x]+F[60,x]
F[73,x] = F[74,x]+F[82,x]
F[74,x] = F[18,x]+F[44,x]+F[75,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[77,x]+F[81,x]
F[77,x] = F[15,x]+F[78,x]
F[78,x] = F[79,x]
F[79,x] = F[4,x]*F[80,x]
F[80,x] = F[4,x]
F[81,x] = F[44,x]
F[82,x] = 2*F[18,x]+F[68,x]+F[83,x]
F[83,x] = F[4,x]*F[84,x]
F[84,x] = F[85,x]+F[86,x]
F[85,x] = F[19,x]
F[86,x] = F[68,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_{26}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \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_{11}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{46}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{38}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{15}\! \left(x \right)
F_{38}\! \left(x \right) = F_{27}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{27}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{27}\! \left(x \right)
F_{46}\! \left(x \right) = F_{18}\! \left(x \right)+F_{47}\! \left(x \right)+F_{70}\! \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_{55}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{7}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{11}\! \left(x \right)
F_{54}\! \left(x \right) = F_{19}\! \left(x \right)
F_{55}\! \left(x \right) = F_{46}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \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_{67}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{65}\! \left(x \right)
F_{63}\! \left(x \right) = F_{11}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{51}\! \left(x \right)
F_{65}\! \left(x \right) = F_{60}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{57}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{60}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{73}\! \left(x \right)
F_{72}\! \left(x \right) = F_{27}\! \left(x \right)+F_{60}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{82}\! \left(x \right)
F_{74}\! \left(x \right) = F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{81}\! \left(x \right)
F_{77}\! \left(x \right) = F_{15}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{4}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right)
F_{81}\! \left(x \right) = F_{44}\! \left(x \right)
F_{82}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{68}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{4}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)+F_{86}\! \left(x \right)
F_{85}\! \left(x \right) = F_{19}\! \left(x \right)
F_{86}\! \left(x \right) = F_{68}\! \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_26(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_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_11(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_15(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_4(x))
Eq(F_26(x), F_27(x) + F_46(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_38(x))
Eq(F_30(x), F_1(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_10(x) + F_34(x))
Eq(F_34(x), F_15(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_15(x))
Eq(F_38(x), F_27(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_27(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_27(x))
Eq(F_46(x), F_18(x) + F_47(x) + F_70(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_55(x))
Eq(F_49(x), F_50(x) + F_7(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_11(x))
Eq(F_54(x), F_19(x))
Eq(F_55(x), F_46(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_67(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_65(x))
Eq(F_63(x), F_11(x) + F_64(x))
Eq(F_64(x), F_51(x))
Eq(F_65(x), F_60(x) + F_66(x))
Eq(F_66(x), F_57(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_60(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_73(x))
Eq(F_72(x), F_27(x) + F_60(x))
Eq(F_73(x), F_74(x) + F_82(x))
Eq(F_74(x), F_18(x) + F_44(x) + F_75(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_77(x) + F_81(x))
Eq(F_77(x), F_15(x) + F_78(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_4(x)*F_80(x))
Eq(F_80(x), F_4(x))
Eq(F_81(x), F_44(x))
Eq(F_82(x), 2*F_18(x) + F_68(x) + F_83(x))
Eq(F_83(x), F_4(x)*F_84(x))
Eq(F_84(x), F_85(x) + F_86(x))
Eq(F_85(x), F_19(x))
Eq(F_86(x), F_68(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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