0132_0231_1023_2031_2130
Counting sequence:
1, 1, 2, 6, 19, 55, 150, 397, 1033, 2656, 6765, 17098, 42934, 107214, 266454, 659441, 1626026, 3996276, 9792873, 23934205, 58356794, 141978223, 344740233, 835552780, 2021764779, 4884476536, 11783809776, 28390825502, 68317920524, 164206869457, 394257850562, 945651195314, 2266051573967, 5425265974611, 12977973172174, 31020415978289, 74090399467049, 176834508849416, 421772964323641, 1005337230369318, 2394857665970890, 5701579594950926, 13566567129948642, 32263760413674241, 76690303419470794, 182203172442345600, 432683525406739045, 1027051243350633817, 2436851484339074098, 5779477827011304531, 13701802204851258505, 32471540754385901908, 76925512046012369559, 182173849760867728884, 431276050303641754724, 1020665747023277937086, 2414768965197041669016, 5711315793951607685601, 13504228058435036534338, 31921301423218822578590, 75434962180799021427019, 178217299575288758404143, 420935722652667811703878, 993971494176039623613525, 2346538120452048846698569, 5538344967108217016157936, 13068752164830899715606341, 30831372752182247189640962, 72720856959337827276938974, 171488557601840331816391502, 404318743769873519014985582, 953076147396439292303827409, 2246201191046564468331200682, 5292846121970671020001445836, 12469573247518126433795421089, 29372281517655198720595816949, 69175056531135181931142986474, 162887972319630386673254710711, 383493491081852592296731249161, 902730729463362921573300135388, 2124669592702670092145018505155, 4999875169748820526842875780976, 11764218447774153306992779379032, 27676086475121785049810219125726, 65100664905895102856301728001444, 153111119471484669966754446756145, 360055930312527752311245975106690, 846596471298527186490918285308362, 1990340884722849676763105107158663, 4678689584265245095261988810755531, 10996841112228571438422678773743870, 25844002352836492341182736203449593, 60729654945414040907290812607144809, 142689545881750265714228421866209304, 335223800269882371587206336551092369, 787462525728621465186941275795584606, 1849602012052738669340563325704843506, 4343911600234023184979728376164790926, 10200909330802000227145784393310733178, 23952613748419605897671308195972240225, 56237229775485173818603774838683793994
Generating function in Maple syntax:
(x^7-3*x^6+8*x^5-12*x^4+16*x^3-14*x^2+6*x-1)/(x-1)/(x^3-2*x^2+3*x-1)^2
Generating function in latex syntax:
\frac{x^{7}-3 x^{6}+8 x^{5}-12 x^{4}+16 x^{3}-14 x^{2}+6 x -1}{\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)^{2}}
Generating function in sympy syntax:
(x**7 - 3*x**6 + 8*x**5 - 12*x**4 + 16*x**3 - 14*x**2 + 6*x - 1)/((x - 1)*(x**3 - 2*x**2 + 3*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^3-2*x^2+3*x-1)^2*F(x)-x^7+3*x^6-8*x^5+12*x^4-16*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)^{2} F \! \left(x \right)-x^{7}+3 x^{6}-8 x^{5}+12 x^{4}-16 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/317400*(-345*(((1/3*n+307/115)*23^(1/2)+I*n-149/15*I)*3^(1/2)+(I*n+921/115*I)*23^(1/2)+n-149/15)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)-2760*(((1/12*n-1/184)*23^(1/2)+I*n+133/24*I)*3^(1/2)+(-1/4*I*n+3/184*I)*23^(1/2)-n-133/24)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)+13800*n+105800)*(11/600*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/317400*(2760*(((-1/12*n+1/184)*23^(1/2)+I*n+133/24*I)*3^(1/2)+(-1/4*I*n+3/184*I)*23^(1/2)+n+133/24)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)+345*(((-1/3*n-307/115)*23^(1/2)+I*n-149/15*I)*3^(1/2)+(I*n+921/115*I)*23^(1/2)-n+149/15)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+13800*n+105800)*(-11/600*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)-1+1/317400*(460*(3^(1/2)*(n-3/46)*23^(1/2)-12*n-133/2)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)+230*(3^(1/2)*(n+921/115)*23^(1/2)+3*n-149/5)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+13800*n+105800)*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-345 \left(\left(\left(\frac{n}{3}+\frac{307}{115}\right) \sqrt{23}+\mathrm{I} n -\frac{149 \,\mathrm{I}}{15}\right) \sqrt{3}+\left(\mathrm{I} n +\frac{921 \,\mathrm{I}}{115}\right) \sqrt{23}+n -\frac{149}{15}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}-2760 \left(\left(\left(\frac{n}{12}-\frac{1}{184}\right) \sqrt{23}+\mathrm{I} n +\frac{133 \,\mathrm{I}}{24}\right) \sqrt{3}+\left(-\frac{\mathrm{I} n}{4}+\frac{3 \,\mathrm{I}}{184}\right) \sqrt{23}-n -\frac{133}{24}\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+13800 n +105800\right) \left(\frac{11 \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}\\+\\\frac{\left(2760 \left(\left(\left(-\frac{n}{12}+\frac{1}{184}\right) \sqrt{23}+\mathrm{I} n +\frac{133 \,\mathrm{I}}{24}\right) \sqrt{3}+\left(-\frac{\mathrm{I} n}{4}+\frac{3 \,\mathrm{I}}{184}\right) \sqrt{23}+n +\frac{133}{24}\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+345 \left(\left(\left(-\frac{n}{3}-\frac{307}{115}\right) \sqrt{23}+\mathrm{I} n -\frac{149 \,\mathrm{I}}{15}\right) \sqrt{3}+\left(\mathrm{I} n +\frac{921 \,\mathrm{I}}{115}\right) \sqrt{23}-n +\frac{149}{15}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+13800 n +105800\right) \left(-\frac{11 \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}\\-1+\\\frac{\left(460 \left(\sqrt{3}\, \left(n -\frac{3}{46}\right) \sqrt{23}-12 n -\frac{133}{2}\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+230 \left(\sqrt{3}\, \left(n +\frac{921}{115}\right) \sqrt{23}+3 n -\frac{149}{5}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+13800 n +105800\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{317400} & \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) = 55
a(6) = 150
a(7) = 397
a(n+6) = -a(n)+4*a(n+1)-10*a(n+2)+14*a(n+3)-13*a(n+4)+6*a(n+5)-1, 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) = 55
a \! \left(6\right) = 150
a \! \left(7\right) = 397
a \! \left(n +6\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right)-1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18806/
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[22,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[20,x]
F[17,x] = F[18,x]+F[4,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[15,x]
F[22,x] = F[2,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[47,x]
F[24,x] = 0
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[32,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[1,x]+F[28,x]
F[31,x] = F[15,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[25,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[46,x]
F[37,x] = F[38,x]+F[44,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[21,x]+F[4,x]
F[42,x] = F[38,x]+F[43,x]
F[43,x] = F[35,x]
F[44,x] = F[45,x]
F[45,x] = F[37,x]*F[4,x]
F[46,x] = F[43,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[2,x]+F[50,x]
F[50,x] = F[24,x]+F[39,x]+F[51,x]
F[51,x] = F[4,x]*F[49,x]
F[52,x] = F[23,x]+F[53,x]
F[53,x] = F[35,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_{22}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{20}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{47}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{32}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{15}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{25}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{46}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{44}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{21}\! \left(x \right)+F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{38}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{35}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{46}\! \left(x \right) = F_{43}\! \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_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{2}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{24}\! \left(x \right)+F_{39}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{23}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{35}\! \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_22(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_20(x))
Eq(F_17(x), F_18(x) + F_4(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_15(x))
Eq(F_22(x), F_2(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_47(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_32(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_1(x) + F_28(x))
Eq(F_31(x), F_15(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_25(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_46(x))
Eq(F_37(x), F_38(x) + F_44(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_21(x) + F_4(x))
Eq(F_42(x), F_38(x) + F_43(x))
Eq(F_43(x), F_35(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_37(x)*F_4(x))
Eq(F_46(x), F_43(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_2(x) + F_50(x))
Eq(F_50(x), F_24(x) + F_39(x) + F_51(x))
Eq(F_51(x), F_4(x)*F_49(x))
Eq(F_52(x), F_23(x) + F_53(x))
Eq(F_53(x), F_35(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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