0132_1032_1203_1230
Counting sequence:
1, 1, 2, 6, 20, 64, 196, 580, 1672, 4724, 13140, 36104, 98244, 265284, 711944, 1901268, 5057396, 13410184, 35468068, 93616612, 246692488, 649213364, 1706714772, 4482989576, 11767419780, 30871698564, 80956727816, 212224977300, 556187966132, 1457311999624, 3817761298660, 10000132645924, 26191226573704, 68591263815284, 179619072094164, 470341114394888, 1231558889913156, 3224653382924484, 8443054093889288, 22105848928539732, 57877241470799348, 151531510480950664, 396728834844144292, 1038678633551479396, 2719355444321916040, 7119486655460768564, 18639306832199900436, 48798847257510976520, 127758079365263161092, 334477114872510859524, 875676783689478300680, 2292560413807830164244, 6002019094432801146164, 15713526705838104603784, 41138621821676482166884, 107702462608181094585508, 281969018204445934337416, 738205105415514228663284, 1932647342877210301607508, 5059739048915140795033352, 13246574127323854359168708, 34679992124082894909680964, 90793420115208151776002312, 237700304538568955534005716, 622307567287475009664221300, 1629222547203651672348765064, 4265360378694757223589675556, 11166859206846546467690240740, 29235218496223479191730556552, 76538798827474573279419551924, 200381183151288581285202551700, 524604761104141804443213421064, 1373433121411786004957841173124, 3595694646222810366615822673284, 9413650904620425027978063300104, 24645258244727207824930062740628, 64522124188461051144905161162676, 168921115047900383971710783656584, 442241222428618443425556496480228, 1157802555222490563478574480839972, 3031166449283482345083312819570568, 7935696804868330395369484371771764, 20775923990104488143125038377221332, 54392075215615555548012741510198536, 142400301758291947348542036828693060, 372808830264777675832100328675930564, 976026189451911562095191385298023176, 2555269738932369380905255732813637460, 6689783029047364134637973691129185012, 17514079351652743757271129230137106824, 45852455032874279858155678045589709220, 120043285761051663764067105533609558884, 314277402278753332337609384877918824584, 822788921132772545075531231882956191284, 2154089361235930666581810056611467426324, 5639479162810228062134121190182279690248, 14764348127670122494906139539496635346436, 38653565221160780893069752521629346746884, 101196347537753310165902594296433231677448, 264935477396020943649094115078550773827092, 693610084658232337032806184698573484837428
Generating function in Maple syntax:
-(2*x^5+7*x^3-12*x^2+6*x-1)/(x^2-3*x+1)/(2*x-1)^2
Generating function in latex syntax:
-\frac{2 x^{5}+7 x^{3}-12 x^{2}+6 x -1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2}}
Generating function in sympy syntax:
(-2*x**5 - 7*x**3 + 12*x**2 - 6*x + 1)/((2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(2*x-1)^2*F(x)+2*x^5+7*x^3-12*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+2 x^{5}+7 x^{3}-12 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/20*(-8*5^(1/2)+40)*(3/2-1/2*5^(1/2))^(-n)+1/20*(8*5^(1/2)+40)*(3/2+1/2*5^(1/2))^(-n)-1/4*2^n*(n+4))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-8 \sqrt{5}+40\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(8 \sqrt{5}+40\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}-\frac{2^{n} \left(n +4\right)}{4} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3), 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) = 20
a \! \left(5\right) = 64
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15498/
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[32,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[26,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]+F[21,x]+F[25,x]
F[20,x] = 0
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[23,x]+F[24,x]
F[23,x] = F[12,x]
F[24,x] = F[19,x]
F[25,x] = F[12,x]*F[14,x]
F[26,x] = F[20,x]+F[27,x]+F[31,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[10,x]
F[30,x] = F[26,x]
F[31,x] = F[12,x]*F[13,x]
F[32,x] = F[33,x]+F[52,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[1,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[36,x]
F[39,x] = F[33,x]+F[40,x]
F[40,x] = F[20,x]+F[41,x]+F[51,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[43,x]+F[46,x]
F[43,x] = F[12,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[12,x]*F[37,x]
F[46,x] = F[47,x]+F[49,x]
F[47,x] = F[20,x]+F[41,x]+F[48,x]
F[48,x] = F[12,x]*F[33,x]
F[49,x] = F[50,x]
F[50,x] = F[12,x]*F[40,x]
F[51,x] = F[12,x]*F[39,x]
F[52,x] = F[20,x]+F[53,x]+F[66,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[55,x]+F[59,x]
F[55,x] = F[10,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[37,x]+F[56,x]
F[59,x] = F[60,x]+F[63,x]
F[60,x] = F[20,x]+F[53,x]+F[61,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[33,x]+F[60,x]
F[63,x] = F[64,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[40,x]+F[63,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[62,x]+F[68,x]
F[68,x] = F[69,x]+F[81,x]
F[69,x] = F[20,x]+F[41,x]+F[70,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[72,x]+F[73,x]
F[72,x] = F[33,x]+F[47,x]
F[73,x] = F[69,x]+F[74,x]
F[74,x] = F[20,x]+F[75,x]+F[76,x]+F[80,x]
F[75,x] = 0
F[76,x] = F[12,x]*F[77,x]
F[77,x] = F[78,x]+F[79,x]
F[78,x] = F[47,x]
F[79,x] = F[74,x]
F[80,x] = F[12,x]*F[69,x]
F[81,x] = F[20,x]+F[82,x]+F[83,x]+F[87,x]
F[82,x] = 0
F[83,x] = F[12,x]*F[84,x]
F[84,x] = F[85,x]+F[86,x]
F[85,x] = F[60,x]
F[86,x] = F[81,x]
F[87,x] = F[12,x]*F[68,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_{32}\! \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_{26}\! \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_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right)
F_{24}\! \left(x \right) = F_{19}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)+F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right)
F_{30}\! \left(x \right) = F_{26}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{52}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{1}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{20}\! \left(x \right)+F_{41}\! \left(x \right)+F_{51}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{46}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{49}\! \left(x \right)
F_{47}\! \left(x \right) = F_{20}\! \left(x \right)+F_{41}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{52}\! \left(x \right) = F_{20}\! \left(x \right)+F_{53}\! \left(x \right)+F_{66}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{59}\! \left(x \right)
F_{55}\! \left(x \right) = F_{10}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{37}\! \left(x \right)+F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{63}\! \left(x \right)
F_{60}\! \left(x \right) = F_{20}\! \left(x \right)+F_{53}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{33}\! \left(x \right)+F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{40}\! \left(x \right)+F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{62}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{81}\! \left(x \right)
F_{69}\! \left(x \right) = F_{20}\! \left(x \right)+F_{41}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{73}\! \left(x \right)
F_{72}\! \left(x \right) = F_{33}\! \left(x \right)+F_{47}\! \left(x \right)
F_{73}\! \left(x \right) = F_{69}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{20}\! \left(x \right)+F_{75}\! \left(x \right)+F_{76}\! \left(x \right)+F_{80}\! \left(x \right)
F_{75}\! \left(x \right) = 0
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{79}\! \left(x \right)
F_{78}\! \left(x \right) = F_{47}\! \left(x \right)
F_{79}\! \left(x \right) = F_{74}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{81}\! \left(x \right) = F_{20}\! \left(x \right)+F_{82}\! \left(x \right)+F_{83}\! \left(x \right)+F_{87}\! \left(x \right)
F_{82}\! \left(x \right) = 0
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)+F_{86}\! \left(x \right)
F_{85}\! \left(x \right) = F_{60}\! \left(x \right)
F_{86}\! \left(x \right) = F_{81}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \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_32(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_26(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_25(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_24(x))
Eq(F_23(x), F_12(x))
Eq(F_24(x), F_19(x))
Eq(F_25(x), F_12(x)*F_14(x))
Eq(F_26(x), F_20(x) + F_27(x) + F_31(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_10(x))
Eq(F_30(x), F_26(x))
Eq(F_31(x), F_12(x)*F_13(x))
Eq(F_32(x), F_33(x) + F_52(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_1(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_36(x))
Eq(F_39(x), F_33(x) + F_40(x))
Eq(F_40(x), F_20(x) + F_41(x) + F_51(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_46(x))
Eq(F_43(x), F_12(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_12(x)*F_37(x))
Eq(F_46(x), F_47(x) + F_49(x))
Eq(F_47(x), F_20(x) + F_41(x) + F_48(x))
Eq(F_48(x), F_12(x)*F_33(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_12(x)*F_40(x))
Eq(F_51(x), F_12(x)*F_39(x))
Eq(F_52(x), F_20(x) + F_53(x) + F_66(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_59(x))
Eq(F_55(x), F_10(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_37(x) + F_56(x))
Eq(F_59(x), F_60(x) + F_63(x))
Eq(F_60(x), F_20(x) + F_53(x) + F_61(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_33(x) + F_60(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_40(x) + F_63(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_62(x) + F_68(x))
Eq(F_68(x), F_69(x) + F_81(x))
Eq(F_69(x), F_20(x) + F_41(x) + F_70(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_73(x))
Eq(F_72(x), F_33(x) + F_47(x))
Eq(F_73(x), F_69(x) + F_74(x))
Eq(F_74(x), F_20(x) + F_75(x) + F_76(x) + F_80(x))
Eq(F_75(x), 0)
Eq(F_76(x), F_12(x)*F_77(x))
Eq(F_77(x), F_78(x) + F_79(x))
Eq(F_78(x), F_47(x))
Eq(F_79(x), F_74(x))
Eq(F_80(x), F_12(x)*F_69(x))
Eq(F_81(x), F_20(x) + F_82(x) + F_83(x) + F_87(x))
Eq(F_82(x), 0)
Eq(F_83(x), F_12(x)*F_84(x))
Eq(F_84(x), F_85(x) + F_86(x))
Eq(F_85(x), F_60(x))
Eq(F_86(x), F_81(x))
Eq(F_87(x), F_12(x)*F_68(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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