0123_0132_0321_1203_1230

Counting sequence:
1, 1, 2, 6, 19, 54, 146, 388, 1023, 2687, 7045, 18456, 48332, 126550, 331329, 867449, 2271031, 5945658, 15565958, 40752232, 106690755, 279320051, 731269417, 1914488220, 5012195264, 13122097594, 34354097541, 89940195053, 235466487643, 616459267902, 1613911316090, 4225274680396, 11061912725127, 28960463495015, 75819477759949, 198497969784864, 519674431594676, 1360525324999198, 3561901543402953, 9325179305209697, 24413636372226175, 63915729811468866, 167333553062180462, 438084929375072560, 1146921235063037259, 3002678775814039259, 7861115092379080561, 20580666501323202468, 53880884411590526888, 141061986733448378242, 369305075788754607885, 966853240632815445461, 2531254646109691728547, 6626910697696259740230, 17349477446979087492194, 45421521643241002736404, 118915087482743920717071, 311323740804990759414863, 815056134932228357527573, 2133844663991694313167912, 5586477857042854581976220, 14625588907136869432760806, 38290288864367753716306257, 100245277685966391716158025, 262445544193531421432167879, 687091354894627872580345674, 1798828520490352196308869206, 4709394206576428716346262008, 12329354099238933952729916883, 32278668091140373141843488707, 84506650174182185472800549305, 221241282431406183276558159276, 579217197120036364356873928592, 1516410308928702909794063626570, 3970013729666072365025316951189, 10393630880069514185281887227069, 27210878910542470190820344730091, 71239005851557896387179146963278, 186506138644131218970717096159818, 488279410080835760524972141516252, 1278332091598376062604199328389015, 3346716864714292427287625843650871, 8761818502544501219258678202563677, 22938738642919211230488408764040240, 60054397426213132472206548089557124, 157224453635720186186131235504631214, 411618963480947426086187158424336601, 1077632436807122092072430239768378673, 2821278346940418850131103560880799503, 7386202604014134458320880442874019922, 19337329465101984524831537767741260350, 50625785791291819116173732860349761216, 132540027908773472823689660813308023387, 346994297935028599354895249579574309035, 908442865896312325240996087925414903809, 2378334299753908376368093014196670402484, 6226560033365412803863282954664596303736, 16301345800342330035221755849797118508818, 42677477367661577301801984594726759222813, 111731086302642401870184197934383159159717, 292515781540265628308750609208422718256435

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

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

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

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

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

Explicit closed form in Maple syntax:
piecewise(n = 0,41,n = 1,15,n = 2,5,n = 3,1,0)-n+2-48/5*(3/2+1/2*5^(1/2))^(-n)*5^(1/2)+48/5*(3/2-1/2*5^(1/2))^(-n)*5^(1/2)-21*(3/2+1/2*5^(1/2))^(-n)-21*(3/2-1/2*5^(1/2))^(-n)

Explicit closed form in latex syntax:
\left(\left\{\begin{array}{cc}41 & n =0 \\ 15 & n =1 \\ 5 & n =2 \\ 1 & n =3 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)-n +2-\frac{48 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}+\frac{48 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}-21 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}-21 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 54
a(6) = 146
a(7) = 388
a(n+2) = -a(n)+3*a(n+1)-1+n, 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) = 19
a \! \left(5\right) = 54
a \! \left(6\right) = 146
a \! \left(7\right) = 388
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-1+n, \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16231/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[14,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[4,x]
F[18,x] = F[12,x]*F[4,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[15,x]+F[21,x]+F[84,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[32,x]
F[23,x] = F[24,x]+F[7,x]
F[24,x] = F[15,x]+F[25,x]+F[28,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[17,x]+F[27,x]
F[27,x] = F[14,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[12,x]
F[31,x] = F[16,x]
F[32,x] = F[20,x]+F[33,x]
F[33,x] = F[15,x]+F[34,x]+F[56,x]+F[57,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[41,x]
F[36,x] = F[24,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[16,x]
F[41,x] = F[33,x]+F[42,x]
F[42,x] = F[15,x]+F[43,x]+F[47,x]+F[48,x]+F[49,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[37,x]
F[46,x] = F[42,x]
F[47,x] = 0
F[48,x] = 0
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[16,x]
F[55,x] = F[52,x]
F[56,x] = 0
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[83,x]
F[59,x] = F[60,x]
F[60,x] = F[15,x]+F[61,x]+F[81,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[66,x]
F[63,x] = F[12,x]+F[64,x]
F[64,x] = F[15,x]+F[25,x]+F[65,x]
F[65,x] = F[30,x]*F[4,x]
F[66,x] = F[60,x]+F[67,x]
F[67,x] = F[15,x]+F[56,x]+F[68,x]+F[80,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[73,x]
F[70,x] = F[64,x]+F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[4,x]*F[40,x]
F[73,x] = F[67,x]+F[74,x]
F[74,x] = F[15,x]+F[47,x]+F[48,x]+F[75,x]+F[79,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[77,x]+F[78,x]
F[77,x] = F[71,x]
F[78,x] = F[74,x]
F[79,x] = F[4,x]*F[51,x]
F[80,x] = F[4,x]*F[59,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[2,x]
F[83,x] = F[52,x]
F[84,x] = F[4,x]*F[85,x]
F[85,x] = F[82,x]+F[86,x]
F[86,x] = F[60,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \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_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{84}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{32}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \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_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right)
F_{31}\! \left(x \right) = F_{16}\! \left(x \right)
F_{32}\! \left(x \right) = F_{20}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{34}\! \left(x \right)+F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{41}\! \left(x \right)
F_{36}\! \left(x \right) = F_{24}\! \left(x \right)+F_{37}\! \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_{40}\! \left(x \right) = F_{16}\! \left(x \right)
F_{41}\! \left(x \right) = F_{33}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{47}\! \left(x \right)+F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{37}\! \left(x \right)
F_{46}\! \left(x \right) = F_{42}\! \left(x \right)
F_{47}\! \left(x \right) = 0
F_{48}\! \left(x \right) = 0
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{16}\! \left(x \right)
F_{55}\! \left(x \right) = F_{52}\! \left(x \right)
F_{56}\! \left(x \right) = 0
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_{83}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{15}\! \left(x \right)+F_{61}\! \left(x \right)+F_{81}\! \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_{66}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{66}\! \left(x \right) = F_{60}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{15}\! \left(x \right)+F_{56}\! \left(x \right)+F_{68}\! \left(x \right)+F_{80}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{73}\! \left(x \right)
F_{70}\! \left(x \right) = F_{64}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{73}\! \left(x \right) = F_{67}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{15}\! \left(x \right)+F_{47}\! \left(x \right)+F_{48}\! \left(x \right)+F_{75}\! \left(x \right)+F_{79}\! \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_{78}\! \left(x \right)
F_{77}\! \left(x \right) = F_{71}\! \left(x \right)
F_{78}\! \left(x \right) = F_{74}\! \left(x \right)
F_{79}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{2}\! \left(x \right)
F_{83}\! \left(x \right) = F_{52}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{82}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{60}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_14(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_12(x)*F_4(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_15(x) + F_21(x) + F_84(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_32(x))
Eq(F_23(x), F_24(x) + F_7(x))
Eq(F_24(x), F_15(x) + F_25(x) + F_28(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_17(x) + F_27(x))
Eq(F_27(x), F_14(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_12(x))
Eq(F_31(x), F_16(x))
Eq(F_32(x), F_20(x) + F_33(x))
Eq(F_33(x), F_15(x) + F_34(x) + F_56(x) + F_57(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_41(x))
Eq(F_36(x), F_24(x) + F_37(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))
Eq(F_40(x), F_16(x))
Eq(F_41(x), F_33(x) + F_42(x))
Eq(F_42(x), F_15(x) + F_43(x) + F_47(x) + F_48(x) + F_49(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_37(x))
Eq(F_46(x), F_42(x))
Eq(F_47(x), 0)
Eq(F_48(x), 0)
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_16(x))
Eq(F_55(x), F_52(x))
Eq(F_56(x), 0)
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_83(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_15(x) + F_61(x) + F_81(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_66(x))
Eq(F_63(x), F_12(x) + F_64(x))
Eq(F_64(x), F_15(x) + F_25(x) + F_65(x))
Eq(F_65(x), F_30(x)*F_4(x))
Eq(F_66(x), F_60(x) + F_67(x))
Eq(F_67(x), F_15(x) + F_56(x) + F_68(x) + F_80(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_73(x))
Eq(F_70(x), F_64(x) + F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_4(x)*F_40(x))
Eq(F_73(x), F_67(x) + F_74(x))
Eq(F_74(x), F_15(x) + F_47(x) + F_48(x) + F_75(x) + F_79(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_77(x) + F_78(x))
Eq(F_77(x), F_71(x))
Eq(F_78(x), F_74(x))
Eq(F_79(x), F_4(x)*F_51(x))
Eq(F_80(x), F_4(x)*F_59(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_2(x))
Eq(F_83(x), F_52(x))
Eq(F_84(x), F_4(x)*F_85(x))
Eq(F_85(x), F_82(x) + F_86(x))
Eq(F_86(x), F_60(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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