0132_0231_0312_2130_3120

Counting sequence:
1, 1, 2, 6, 19, 54, 137, 319, 700, 1477, 3040, 6164, 12393, 24808, 49563, 98957, 197578, 394591, 788314, 1575370, 3148991, 6295626, 12588157, 25172331, 50339624, 100672969, 201338212, 402667024, 805322725, 1610631932, 3221247855, 6442476889, 12884931798, 25769838083, 51539646718, 103079259622, 206158480603, 412316917246, 824633784689, 1649267513175, 3298534963156, 6597069855501, 13194139631912, 26388279175756, 52776558253729, 105553116399184, 211106232678787, 422212465225829, 844424930306850, 1688849860454887, 3377699720735970, 6755399441282114, 13510798882357303, 27021597764489458, 54043195528734373, 108086391057203587, 216172782114120128, 432345564227930001, 864691128455525164, 1729382256910689480, 3458764513820990621, 6917529027641563876, 13835058055282679767, 27670116110564879281, 55340232221129244334, 110680464442257938699, 221360928884515289862, 442721857769029952734, 885443715538059237075, 1770887431076117762342, 3541774862152234767385, 7083549724304468729839, 14167099448608936604908, 28334198897217872302933, 56668397794435743644528, 113336795588871486270852, 226673591177742971464153, 453347182355485941788856, 906694364710971882373739, 1813388729421943763476285, 3626777458843887525611386, 7253554917687775049808751, 14507109835375550098127722, 29014219670751100194686906, 58028439341502200387723439, 116056878683004400773711514, 232113757366008801545599437, 464227514732017603089283739, 928455029464035206176557400, 1856910058928070412351006297, 3713820117856140824699802100, 7427640235712281649397288064, 14855280471424563298792150613, 29710560942849126597581762508, 59421121885698253195160869183, 118842243771396506390318961417, 237684487542793012780635020678, 475368975085586025561267009811, 950737950171172051122530854414, 1901475900342344102245058405590, 3802951800684688204490113365451

Generating function in Maple syntax:
-(5*x^5-12*x^4+18*x^3-15*x^2+6*x-1)/(2*x-1)/(x-1)^5

Generating function in latex syntax:
-\frac{5 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1}{\left(2 x -1\right) \left(x -1\right)^{5}}

Generating function in sympy syntax:
(-5*x**5 + 12*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))

Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^5*F(x)+5*x^5-12*x^4+18*x^3-15*x^2+6*x-1 = 0

Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+5 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0

Explicit closed form in Maple syntax:
-2-1/24*n^2-31/12*n-5/12*n^3+1/24*n^4+3*2^n

Explicit closed form in latex syntax:
-2-\frac{n^{2}}{24}-\frac{31 n}{12}-\frac{5 n^{3}}{12}+\frac{n^{4}}{24}+3 \,2^{n}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 54
a(n+1) = 2*a(n)-1/24*(n-1)*(n^3-13*n^2+10*n-24), n >= 6

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) = 54
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(n^{3}-13 n^{2}+10 n -24\right)}{24}, \quad n \geq 6

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18630/
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[20,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[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[10,x]+F[17,x]
F[20,x] = F[21,x]+F[30,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[29,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[9,x]
F[28,x] = F[14,x]
F[29,x] = F[21,x]
F[30,x] = F[31,x]+F[32,x]+F[52,x]
F[31,x] = 0
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[34,x]+F[51,x]
F[34,x] = F[35,x]+F[6,x]
F[35,x] = F[31,x]+F[36,x]+F[42,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[10,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[38,x]
F[41,x] = F[17,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[13,x]+F[44,x]
F[44,x] = F[45,x]+F[48,x]
F[45,x] = F[46,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[14,x]+F[45,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[17,x]+F[48,x]
F[51,x] = F[30,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]+F[63,x]
F[54,x] = F[21,x]+F[55,x]
F[55,x] = F[31,x]+F[56,x]+F[62,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]+F[61,x]
F[58,x] = F[10,x]+F[59,x]
F[59,x] = F[31,x]+F[36,x]+F[60,x]
F[60,x] = F[12,x]*F[13,x]
F[61,x] = F[55,x]
F[62,x] = F[12,x]*F[54,x]
F[63,x] = F[64,x]+F[76,x]
F[64,x] = F[31,x]+F[65,x]+F[74,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[14,x]+F[68,x]
F[68,x] = F[31,x]+F[46,x]+F[69,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[12,x]+F[72,x]
F[72,x] = F[73,x]
F[73,x] = F[12,x]*F[71,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[21,x]+F[64,x]
F[76,x] = 2*F[31,x]+F[77,x]+F[88,x]
F[77,x] = F[12,x]*F[78,x]
F[78,x] = F[79,x]
F[79,x] = F[17,x]+F[80,x]
F[80,x] = 2*F[31,x]+F[49,x]+F[81,x]
F[81,x] = F[12,x]*F[82,x]
F[82,x] = F[83,x]
F[83,x] = F[84,x]+F[86,x]
F[84,x] = F[85,x]
F[85,x] = F[10,x]*F[12,x]
F[86,x] = F[87,x]
F[87,x] = F[12,x]*F[83,x]
F[88,x] = F[12,x]*F[89,x]
F[89,x] = F[55,x]+F[76,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_{20}\! \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_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \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_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{30}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{29}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{9}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14}\! \left(x \right)
F_{29}\! \left(x \right) = F_{21}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{52}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{51}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{6}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)+F_{36}\! \left(x \right)+F_{42}\! \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_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{17}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{13}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{14}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{17}\! \left(x \right)+F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{30}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{63}\! \left(x \right)
F_{54}\! \left(x \right) = F_{21}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{31}\! \left(x \right)+F_{56}\! \left(x \right)+F_{62}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{61}\! \left(x \right)
F_{58}\! \left(x \right) = F_{10}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{31}\! \left(x \right)+F_{36}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{61}\! \left(x \right) = F_{55}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{76}\! \left(x \right)
F_{64}\! \left(x \right) = F_{31}\! \left(x \right)+F_{65}\! \left(x \right)+F_{74}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{14}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{31}\! \left(x \right)+F_{46}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{21}\! \left(x \right)+F_{64}\! \left(x \right)
F_{76}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{77}\! \left(x \right)+F_{88}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{17}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{49}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{86}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right) F_{83}\! \left(x \right)
F_{88}\! \left(x \right) = F_{12}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{55}\! \left(x \right)+F_{76}\! \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_20(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_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_10(x) + F_17(x))
Eq(F_20(x), F_21(x) + F_30(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_29(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_9(x))
Eq(F_28(x), F_14(x))
Eq(F_29(x), F_21(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_52(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_51(x))
Eq(F_34(x), F_35(x) + F_6(x))
Eq(F_35(x), F_31(x) + F_36(x) + F_42(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_10(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x)*F_38(x))
Eq(F_41(x), F_17(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_13(x) + F_44(x))
Eq(F_44(x), F_45(x) + F_48(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_14(x) + F_45(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_17(x) + F_48(x))
Eq(F_51(x), F_30(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_63(x))
Eq(F_54(x), F_21(x) + F_55(x))
Eq(F_55(x), F_31(x) + F_56(x) + F_62(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_61(x))
Eq(F_58(x), F_10(x) + F_59(x))
Eq(F_59(x), F_31(x) + F_36(x) + F_60(x))
Eq(F_60(x), F_12(x)*F_13(x))
Eq(F_61(x), F_55(x))
Eq(F_62(x), F_12(x)*F_54(x))
Eq(F_63(x), F_64(x) + F_76(x))
Eq(F_64(x), F_31(x) + F_65(x) + F_74(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_14(x) + F_68(x))
Eq(F_68(x), F_31(x) + F_46(x) + F_69(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_12(x) + F_72(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_12(x)*F_71(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_21(x) + F_64(x))
Eq(F_76(x), 2*F_31(x) + F_77(x) + F_88(x))
Eq(F_77(x), F_12(x)*F_78(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_17(x) + F_80(x))
Eq(F_80(x), 2*F_31(x) + F_49(x) + F_81(x))
Eq(F_81(x), F_12(x)*F_82(x))
Eq(F_82(x), F_83(x))
Eq(F_83(x), F_84(x) + F_86(x))
Eq(F_84(x), F_85(x))
Eq(F_85(x), F_10(x)*F_12(x))
Eq(F_86(x), F_87(x))
Eq(F_87(x), F_12(x)*F_83(x))
Eq(F_88(x), F_12(x)*F_89(x))
Eq(F_89(x), F_55(x) + F_76(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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