0132_0231_0312_1032_3120
Counting sequence:
1, 1, 2, 6, 19, 56, 152, 389, 953, 2261, 5237, 11909, 26693, 59141, 129797, 282629, 611333, 1314821, 2813957, 5996549, 12730373, 26935301, 56819717, 119537669, 250871813, 525336581, 1097859077, 2290089989, 4768923653, 9915334661, 20585644037, 42681237509, 88382373893, 182804545541, 377688686597, 779536564229, 1607391510533, 3311419785221, 6816113098757, 14018773254149, 28810640621573, 59167469469701, 121427315392517, 249039383691269, 510448273195013, 1045635558014981, 2140749139279877, 4380454325059589, 8958820743118853, 18313465672237061, 37418579716472837, 76420456176943109, 156007505841881093, 318348198659751941, 649362771271483397, 1324058290446925829, 2698782076701769733, 5498895145019375621, 11200452273270423557, 22806228513004191749, 46423104958935072773, 94467505783723524101, 192177603299153805317, 390840390061721124869, 794651147050269278213, 1615243027954192613381, 3282367523615693340677, 6668497982646002909189, 13544521836121238274053, 27504095413900941459461, 55838294311118812741637, 113336795588871485128709, 229994005111010689548293, 466628838088556817678341, 946539331910184512520197, 1919641975286510779367429, 3892410573505305067388933, 7891074392875177152086021, 15994655277479488338788357, 32414323538417244746809349, 65678673043751025632083973, 133057398021335123541098501, 269514899910336391636058117, 545830007556005072379838469, 1105260430582674722975121413, 2237721692106678602381131781, 4529845046096015517624041477, 9168493415957347660971638789, 18554593479445328573390389253, 37544400253951923649675001861, 75959227098026380305138450437, 153659307376297826621853794309, 310800321113085785266861375493, 628564054947151834580030324741, 1271054935336264197252675796997, 2569963521556449450690581889029, 5195634344880741013751624368133, 10502683293297166252244169916421, 21228195793665700953970182193157, 42902050001474138806904049106949, 86695416831233751411735467655173
Generating function in Maple syntax:
-(x^7+x^5+x^4+5*x^2-4*x+1)/(x-1)/(2*x-1)^2
Generating function in latex syntax:
-\frac{x^{7}+x^{5}+x^{4}+5 x^{2}-4 x +1}{\left(x -1\right) \left(2 x -1\right)^{2}}
Generating function in sympy syntax:
(-x**7 - x**5 - x**4 - 5*x**2 + 4*x - 1)/((x - 1)*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x-1)^2*F(x)+x^7+x^5+x^4+5*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+x^{7}+x^{5}+x^{4}+5 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,n = 4,19,5+1/64*(45*n-123)*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ 19 & n =4 \\ 5+\frac{\left(45 n -123\right) 2^{n}}{64} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(6) = 152
a(7) = 389
a(n+2) = -4*a(n)+4*a(n+1)+5, 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) = 56
a \! \left(6\right) = 152
a \! \left(7\right) = 389
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+5, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21651/
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[13,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[8,x]+F[80,x]
F[8,x] = F[65,x]+F[9,x]
F[9,x] = F[10,x]+F[22,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[14,x]
F[13,x] = x
F[14,x] = F[1,x]+F[15,x]+F[17,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = 2*F[15,x]+F[1,x]
F[17,x] = F[18,x]
F[18,x] = F[13,x]*F[16,x]*F[19,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[13,x]*F[19,x]
F[22,x] = F[13,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[26,x]
F[24,x] = 0
F[25,x] = F[13,x]*F[20,x]
F[26,x] = F[13,x]*F[27,x]
F[27,x] = F[28,x]+F[43,x]
F[28,x] = F[13,x]+F[29,x]
F[29,x] = F[24,x]+F[30,x]+F[31,x]
F[30,x] = x^2
F[31,x] = F[13,x]*F[32,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[13,x]+F[34,x]
F[34,x] = F[24,x]+F[30,x]+F[35,x]
F[35,x] = F[13,x]*F[33,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[24,x]+F[30,x]+F[38,x]
F[38,x] = F[13,x]*F[39,x]
F[39,x] = F[13,x]+F[37,x]
F[40,x] = F[41,x]
F[41,x] = F[13,x]*F[42,x]
F[42,x] = F[34,x]+F[40,x]
F[43,x] = F[44,x]+F[47,x]
F[44,x] = F[24,x]+F[25,x]+F[45,x]
F[45,x] = F[13,x]*F[46,x]
F[46,x] = F[13,x]+F[44,x]
F[47,x] = F[24,x]+F[48,x]+F[51,x]+F[56,x]
F[48,x] = F[13,x]*F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[13,x]*F[20,x]
F[51,x] = F[13,x]*F[52,x]
F[52,x] = F[34,x]+F[53,x]
F[53,x] = F[24,x]+F[48,x]+F[51,x]+F[54,x]
F[54,x] = F[13,x]*F[55,x]
F[55,x] = F[44,x]+F[53,x]
F[56,x] = F[13,x]*F[57,x]
F[57,x] = F[55,x]+F[58,x]
F[58,x] = F[59,x]+F[62,x]
F[59,x] = 2*F[24,x]+F[48,x]+F[60,x]
F[60,x] = F[13,x]*F[61,x]
F[61,x] = F[44,x]+F[59,x]
F[62,x] = F[63,x]
F[63,x] = F[13,x]*F[64,x]
F[64,x] = F[53,x]+F[62,x]
F[65,x] = F[66,x]
F[66,x] = F[67,x]*F[73,x]
F[67,x] = F[68,x]+F[70,x]
F[68,x] = F[19,x]*F[69,x]
F[69,x] = F[1,x]+F[13,x]
F[70,x] = F[71,x]
F[71,x] = F[13,x]*F[19,x]*F[72,x]
F[72,x] = F[11,x]+F[25,x]
F[73,x] = F[74,x]
F[74,x] = F[13,x]*F[75,x]
F[75,x] = F[1,x]+F[76,x]+F[78,x]
F[76,x] = F[13,x]*F[77,x]
F[77,x] = 2*F[76,x]+F[1,x]
F[78,x] = F[79,x]
F[79,x] = F[13,x]*F[19,x]*F[77,x]
F[80,x] = F[81,x]
F[81,x] = F[82,x]+F[84,x]
F[82,x] = F[11,x]*F[69,x]*F[83,x]
F[83,x] = F[1,x]+F[73,x]
F[84,x] = F[13,x]*F[20,x]*F[85,x]
F[85,x] = F[13,x]+F[83,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)+F_{80}\! \left(x \right)
F_{8}\! \left(x \right) = F_{65}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{22}\! \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_{14}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{1}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{43}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{24}\! \left(x \right)+F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = x^{2}
F_{31}\! \left(x \right) = F_{13}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{24}\! \left(x \right)+F_{30}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{13}\! \left(x \right) F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{24}\! \left(x \right)+F_{30}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{13}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{13}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{34}\! \left(x \right)+F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{13}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{13}\! \left(x \right)+F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{24}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)+F_{56}\! \left(x \right)
F_{48}\! \left(x \right) = F_{13}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{51}\! \left(x \right) = F_{13}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{34}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{24}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{13}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{44}\! \left(x \right)+F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{13}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{55}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{62}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{48}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{13}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{44}\! \left(x \right)+F_{59}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{13}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{53}\! \left(x \right)+F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right) F_{73}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{70}\! \left(x \right)
F_{68}\! \left(x \right) = F_{19}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{13}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{1}\! \left(x \right)+F_{76}\! \left(x \right)+F_{78}\! \left(x \right)
F_{76}\! \left(x \right) = F_{13}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = 2 F_{76}\! \left(x \right)+F_{1}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right) F_{77}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{84}\! \left(x \right)
F_{82}\! \left(x \right) = F_{11}\! \left(x \right) F_{69}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{1}\! \left(x \right)+F_{73}\! \left(x \right)
F_{84}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{13}\! \left(x \right)+F_{83}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_13(x)*F_7(x))
Eq(F_7(x), F_8(x) + F_80(x))
Eq(F_8(x), F_65(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_22(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_14(x))
Eq(F_13(x), x)
Eq(F_14(x), F_1(x) + F_15(x) + F_17(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), F_1(x) + 2*F_15(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_13(x)*F_16(x)*F_19(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_13(x)*F_19(x))
Eq(F_22(x), F_13(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_26(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_13(x)*F_20(x))
Eq(F_26(x), F_13(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_43(x))
Eq(F_28(x), F_13(x) + F_29(x))
Eq(F_29(x), F_24(x) + F_30(x) + F_31(x))
Eq(F_30(x), x**2)
Eq(F_31(x), F_13(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_13(x) + F_34(x))
Eq(F_34(x), F_24(x) + F_30(x) + F_35(x))
Eq(F_35(x), F_13(x)*F_33(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_24(x) + F_30(x) + F_38(x))
Eq(F_38(x), F_13(x)*F_39(x))
Eq(F_39(x), F_13(x) + F_37(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_13(x)*F_42(x))
Eq(F_42(x), F_34(x) + F_40(x))
Eq(F_43(x), F_44(x) + F_47(x))
Eq(F_44(x), F_24(x) + F_25(x) + F_45(x))
Eq(F_45(x), F_13(x)*F_46(x))
Eq(F_46(x), F_13(x) + F_44(x))
Eq(F_47(x), F_24(x) + F_48(x) + F_51(x) + F_56(x))
Eq(F_48(x), F_13(x)*F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_13(x)*F_20(x))
Eq(F_51(x), F_13(x)*F_52(x))
Eq(F_52(x), F_34(x) + F_53(x))
Eq(F_53(x), F_24(x) + F_48(x) + F_51(x) + F_54(x))
Eq(F_54(x), F_13(x)*F_55(x))
Eq(F_55(x), F_44(x) + F_53(x))
Eq(F_56(x), F_13(x)*F_57(x))
Eq(F_57(x), F_55(x) + F_58(x))
Eq(F_58(x), F_59(x) + F_62(x))
Eq(F_59(x), 2*F_24(x) + F_48(x) + F_60(x))
Eq(F_60(x), F_13(x)*F_61(x))
Eq(F_61(x), F_44(x) + F_59(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_13(x)*F_64(x))
Eq(F_64(x), F_53(x) + F_62(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_67(x)*F_73(x))
Eq(F_67(x), F_68(x) + F_70(x))
Eq(F_68(x), F_19(x)*F_69(x))
Eq(F_69(x), F_1(x) + F_13(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_13(x)*F_19(x)*F_72(x))
Eq(F_72(x), F_11(x) + F_25(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_13(x)*F_75(x))
Eq(F_75(x), F_1(x) + F_76(x) + F_78(x))
Eq(F_76(x), F_13(x)*F_77(x))
Eq(F_77(x), F_1(x) + 2*F_76(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_13(x)*F_19(x)*F_77(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_82(x) + F_84(x))
Eq(F_82(x), F_11(x)*F_69(x)*F_83(x))
Eq(F_83(x), F_1(x) + F_73(x))
Eq(F_84(x), F_13(x)*F_20(x)*F_85(x))
Eq(F_85(x), F_13(x) + F_83(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": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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"pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 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