0123_0231_0321_1032_1230
Counting sequence:
1, 1, 2, 6, 19, 53, 147, 400, 1075, 2859, 7539, 19738, 51363, 132961, 342627, 879396, 2249123, 5734231, 14578467, 36969518, 93535091, 236153229, 595086195, 1496928440, 3759393363, 9427218563, 23607287571, 59040211266, 147478002307, 367975217849, 917176734147, 2283804280396, 5681484779587, 14121648403119, 35071230197571, 87031880585302, 215816983554003, 534797603852005, 1324357695761235, 3277535760539616, 8106420251673971, 20038381098719707, 49506183153611187, 122244753649769834, 301706703645304163, 744274193568512529, 1835198169230751267, 4523172721554993524, 11143491069839117283, 27442551965754964487, 67555336667890898019, 166239105739142200446, 408932050687928030515, 1005586092636109164605, 2471968509544120897587, 6074734544413410938632, 14923724737333037270419, 36651969729692734449459, 89989522756907098604883, 220884518074497365500306, 542027423128071289723203, 1329730595605969299023209, 3261339941112838056040323, 7996904362652631101721756, 19603987762832899808990339, 48047051965969015509333343, 117731274946486899956425347, 288418140440025338469351590, 706417816240418592118231955, 1729862832329707076199189909, 4235211860131402866726464019, 10367032370464497607565196848, 25371836616035937799892867763, 62082571450359437441336031371, 151883271227376480868571138547, 371513563174178799784475823546, 908585587824488549835524023843, 2221699568589731238857523862721, 5431689574785856175752572970467, 13277503247491829226168672237508, 32451249978212191047903923534371, 79301535550131949797410533917879, 193761939679350244013407026682275, 473362184456796483041022672518350, 1156267466289745453899730577502963, 2823996201977855923665838324328173, 6896239197825396610686394425551091, 16838532337730096296773956071017816, 41109398681068422817159951558153427, 100351577055534056308680478064045539, 244937142311253597802619790430252947, 597769288071942491027704443289289186, 1458687160762058120453696328482315267, 3559109920603799052240216787565625369, 8683051066292056406140036929710461251, 21181466492840452547237224386492044012, 51664636995600443047254260207802436803, 126004300810948842417742227551818188815, 307279012214940616981371455315989244355, 749267432762622558353747100942618808566, 1826829866381213186734022322723421553491
Generating function in Maple syntax:
-(x^6+2*x^5-4*x^4-4*x^3+3*x-1)/(x^2+2*x-1)^2
Generating function in latex syntax:
-\frac{x^{6}+2 x^{5}-4 x^{4}-4 x^{3}+3 x -1}{\left(x^{2}+2 x -1\right)^{2}}
Generating function in sympy syntax:
(-x**6 - 2*x**5 + 4*x**4 + 4*x**3 - 3*x + 1)/(x**2 + 2*x - 1)**2
Implicit equation for the generating function in Maple syntax:
(x^2+2*x-1)^2*F(x)+x^6+2*x^5-4*x^4-4*x^3+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+2 x -1\right)^{2} F \! \left(x \right)+x^{6}+2 x^{5}-4 x^{4}-4 x^{3}+3 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/16*((6*n+15)*2^(1/2)+10*n+24)*(-1-2^(1/2))^(-n)-3/8*(2^(1/2)-1)^(-n)*((n+5/2)*2^(1/2)-5/3*n-4))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(\left(6 n +15\right) \sqrt{2}+10 n +24\right) \left(-1-\sqrt{2}\right)^{-n}}{16}-\\\frac{3 \left(\sqrt{2}-1\right)^{-n} \left(\left(n +\frac{5}{2}\right) \sqrt{2}-\frac{5 n}{3}-4\right)}{8} & \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) = 53
a(6) = 147
a(n+4) = -a(n)-4*a(n+1)-2*a(n+2)+4*a(n+3), n >= 7
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) = 53
a \! \left(6\right) = 147
a \! \left(n +4\right) = -a \! \left(n \right)-4 a \! \left(n +1\right)-2 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17072/
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[20,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[18,x] = F[19,x]*F[4,x]
F[19,x] = F[4,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[83,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[36,x]
F[25,x] = F[15,x]+F[26,x]
F[26,x] = F[22,x]+F[27,x]+F[34,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[30,x]+F[4,x]
F[30,x] = F[31,x]
F[31,x] = x^2
F[32,x] = F[33,x]
F[33,x] = F[18,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[15,x]+F[33,x]
F[36,x] = F[37,x]+F[68,x]
F[37,x] = F[22,x]+F[23,x]+F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[2,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[49,x]
F[43,x] = F[4,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[4,x]
F[48,x] = F[33,x]
F[49,x] = F[40,x]+F[50,x]
F[50,x] = 2*F[22,x]+F[51,x]+F[64,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[57,x]
F[53,x] = F[33,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[33,x]
F[57,x] = F[58,x]+F[61,x]
F[58,x] = 2*F[22,x]+F[51,x]+F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[40,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[58,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[67,x]
F[66,x] = F[40,x]
F[67,x] = F[58,x]
F[68,x] = F[22,x]+F[69,x]+F[77,x]+F[81,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[71,x]+F[74,x]
F[71,x] = F[72,x]+F[73,x]
F[72,x] = F[18,x]
F[73,x] = F[55,x]
F[74,x] = F[75,x]+F[76,x]
F[75,x] = 2*F[22,x]+F[59,x]+F[69,x]
F[76,x] = F[62,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[79,x]+F[80,x]
F[79,x] = F[40,x]
F[80,x] = F[58,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[37,x]+F[58,x]
F[83,x] = F[4,x]*F[84,x]
F[84,x] = F[85,x]+F[87,x]
F[85,x] = F[2,x]+F[86,x]
F[86,x] = F[41,x]
F[87,x] = F[37,x]+F[88,x]
F[88,x] = 2*F[22,x]+F[51,x]+F[59,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_{20}\! \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_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{83}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{36}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{34}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = x^{2}
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{68}\! \left(x \right)
F_{37}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \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_{2}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \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_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right)
F_{48}\! \left(x \right) = F_{33}\! \left(x \right)
F_{49}\! \left(x \right) = F_{40}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{51}\! \left(x \right)+F_{64}\! \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_{57}\! \left(x \right)
F_{53}\! \left(x \right) = F_{33}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{33}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{61}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{51}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{40}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{58}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{67}\! \left(x \right)
F_{66}\! \left(x \right) = F_{40}\! \left(x \right)
F_{67}\! \left(x \right) = F_{58}\! \left(x \right)
F_{68}\! \left(x \right) = F_{22}\! \left(x \right)+F_{69}\! \left(x \right)+F_{77}\! \left(x \right)+F_{81}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{74}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{73}\! \left(x \right)
F_{72}\! \left(x \right) = F_{18}\! \left(x \right)
F_{73}\! \left(x \right) = F_{55}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{76}\! \left(x \right)
F_{75}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{59}\! \left(x \right)+F_{69}\! \left(x \right)
F_{76}\! \left(x \right) = F_{62}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{80}\! \left(x \right)
F_{79}\! \left(x \right) = F_{40}\! \left(x \right)
F_{80}\! \left(x \right) = F_{58}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{37}\! \left(x \right)+F_{58}\! \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_{87}\! \left(x \right)
F_{85}\! \left(x \right) = F_{2}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{41}\! \left(x \right)
F_{87}\! \left(x \right) = F_{37}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{51}\! \left(x \right)+F_{59}\! \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_20(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))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_4(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_83(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_36(x))
Eq(F_25(x), F_15(x) + F_26(x))
Eq(F_26(x), F_22(x) + F_27(x) + F_34(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_30(x) + F_4(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), x**2)
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_18(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_15(x) + F_33(x))
Eq(F_36(x), F_37(x) + F_68(x))
Eq(F_37(x), F_22(x) + F_23(x) + F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_2(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_49(x))
Eq(F_43(x), F_4(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_4(x))
Eq(F_48(x), F_33(x))
Eq(F_49(x), F_40(x) + F_50(x))
Eq(F_50(x), 2*F_22(x) + F_51(x) + F_64(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_57(x))
Eq(F_53(x), F_33(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_33(x))
Eq(F_57(x), F_58(x) + F_61(x))
Eq(F_58(x), 2*F_22(x) + F_51(x) + F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_40(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_58(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_67(x))
Eq(F_66(x), F_40(x))
Eq(F_67(x), F_58(x))
Eq(F_68(x), F_22(x) + F_69(x) + F_77(x) + F_81(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_71(x) + F_74(x))
Eq(F_71(x), F_72(x) + F_73(x))
Eq(F_72(x), F_18(x))
Eq(F_73(x), F_55(x))
Eq(F_74(x), F_75(x) + F_76(x))
Eq(F_75(x), 2*F_22(x) + F_59(x) + F_69(x))
Eq(F_76(x), F_62(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_79(x) + F_80(x))
Eq(F_79(x), F_40(x))
Eq(F_80(x), F_58(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_37(x) + F_58(x))
Eq(F_83(x), F_4(x)*F_84(x))
Eq(F_84(x), F_85(x) + F_87(x))
Eq(F_85(x), F_2(x) + F_86(x))
Eq(F_86(x), F_41(x))
Eq(F_87(x), F_37(x) + F_88(x))
Eq(F_88(x), 2*F_22(x) + F_51(x) + F_59(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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