0213_1230_2301_3021
Counting sequence:
1, 1, 2, 6, 20, 60, 156, 369, 821, 1755, 3655, 7486, 15174, 30566, 61350, 122895, 245931, 491909, 983721, 1967140, 3933700, 7866456, 15731504, 31461021, 62919345, 125835135, 251665691, 503325594, 1006643986, 2013279130, 4026547530, 8053082171, 16106148999, 32212279881, 64424538525, 128849052320, 257698076016, 515396119084, 1030792200436, 2061584357865, 4123168666925, 8246337278691, 16492674495279, 32985348920886, 65970697763870, 131941395440910, 263882790785326, 527765581463719, 1055531162809251, 2111062325488205, 4222124650833105, 8444249301508956, 16888498602845724, 33776997205503296, 67553994410801400, 135107988821379445, 270215977642516201, 540431955284769159, 1080863910569253251, 2161727821138198290, 4323455642276063850, 8646911284551769026, 17293822569103151954, 34587645138205888851, 69175290276411332095, 138350580552822186385, 276701161105643861061, 553402322211287174744, 1106804644422573764616, 2213609288845146904980, 4427218577690293144380, 8854437155380585579841, 17708874310761170405349, 35417748621522340008811, 70835497243044679165975, 141670994486089357428270, 283341988972178713898486, 566683977944357426782134, 1133367955888714852490166, 2266735911777429703844415, 4533471823554859406488475, 9066943647109718811709461, 18133887294219437622081529, 36267774588438875242752916, 72535549176877750484020020, 145071098353755500966475560, 290142196707511001931304896, 580284393415022003860878669, 1160568786830044007719938081, 2321137573660088015437965455, 4642275147320176030873925355, 9284550294640352061745746826, 18569100589280704123489287874, 37138201178561408246976264426, 74276402357122816493950108250, 148552804714245632987897682795, 297105609428491265975792714871, 594211218856982531951582658009, 1188422437713965063903162419181, 2376844875427930127806321812240, 4753689750855860255612640464800
Generating function in Maple syntax:
-(4*x^8-7*x^7+3*x^6+6*x^5-13*x^4+18*x^3-15*x^2+6*x-1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
-\frac{4 x^{8}-7 x^{7}+3 x^{6}+6 x^{5}-13 x^{4}+18 x^{3}-15 x^{2}+6 x -1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-4*x**8 + 7*x**7 - 3*x**6 - 6*x**5 + 13*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^5*F(x)+4*x^8-7*x^7+3*x^6+6*x^5-13*x^4+18*x^3-15*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+4 x^{8}-7 x^{7}+3 x^{6}+6 x^{5}-13 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,15+15/4*2^n-65/4*n+71/24*n^2-3/4*n^3+1/24*n^4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 15+\frac{15 \,2^{n}}{4}-\frac{65 n}{4}+\frac{71 n^{2}}{24}-\frac{3 n^{3}}{4}+\frac{n^{4}}{24} & \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) = 60
a(6) = 156
a(7) = 369
a(8) = 821
a(n+1) = -1/24*n^4+11/12*n^3-119/24*n^2+2*a(n)+241/12*n-29, n >= 9
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) = 60
a \! \left(6\right) = 156
a \! \left(7\right) = 369
a \! \left(8\right) = 821
a \! \left(n +1\right) = -\frac{n^{4}}{24}+\frac{11 n^{3}}{12}-\frac{119 n^{2}}{24}+2 a \! \left(n \right)+\frac{241 n}{12}-29, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15763/
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[16,x]*F[4,x]
F[4,x] = F[26,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[16,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[21,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[16,x]
F[12,x] = F[13,x]+F[17,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = x
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[16,x]*F[20,x]
F[20,x] = F[14,x]
F[21,x] = F[22,x]
F[22,x] = F[16,x]*F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[14,x]+F[25,x]
F[25,x] = F[22,x]
F[26,x] = F[27,x]+F[48,x]
F[27,x] = F[28,x]
F[28,x] = F[16,x]*F[29,x]
F[29,x] = F[13,x]+F[30,x]
F[30,x] = F[27,x]+F[31,x]
F[31,x] = F[32,x]+F[33,x]+F[43,x]
F[32,x] = 0
F[33,x] = F[16,x]*F[34,x]
F[34,x] = F[35,x]+F[37,x]
F[35,x] = F[14,x]+F[36,x]
F[36,x] = F[19,x]
F[37,x] = F[31,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[16,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[36,x]
F[42,x] = F[38,x]
F[43,x] = F[16,x]*F[44,x]
F[44,x] = F[14,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[16,x]*F[47,x]
F[47,x] = F[14,x]+F[45,x]
F[48,x] = F[32,x]+F[49,x]+F[79,x]
F[49,x] = F[16,x]*F[50,x]
F[50,x] = F[51,x]+F[60,x]
F[51,x] = F[52,x]+F[6,x]
F[52,x] = F[32,x]+F[53,x]+F[55,x]
F[53,x] = F[16,x]*F[54,x]
F[54,x] = F[25,x]+F[6,x]
F[55,x] = F[16,x]*F[56,x]
F[56,x] = F[24,x]+F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[16,x]*F[59,x]
F[59,x] = F[14,x]
F[60,x] = F[48,x]+F[61,x]
F[61,x] = F[32,x]+F[62,x]+F[74,x]+F[88,x]
F[62,x] = F[16,x]*F[63,x]
F[63,x] = F[64,x]+F[68,x]
F[64,x] = F[52,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[16,x]*F[67,x]
F[67,x] = F[25,x]
F[68,x] = F[61,x]+F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[16,x]*F[71,x]
F[71,x] = F[72,x]+F[73,x]
F[72,x] = F[65,x]
F[73,x] = F[69,x]
F[74,x] = F[16,x]*F[75,x]
F[75,x] = F[76,x]+F[87,x]
F[76,x] = F[32,x]+F[77,x]+F[79,x]
F[77,x] = F[16,x]*F[78,x]
F[78,x] = F[6,x]+F[76,x]
F[79,x] = F[16,x]*F[80,x]
F[80,x] = F[81,x]+F[82,x]
F[81,x] = F[14,x]+F[76,x]
F[82,x] = F[83,x]+F[84,x]
F[83,x] = F[46,x]
F[84,x] = F[85,x]
F[85,x] = F[16,x]*F[86,x]
F[86,x] = F[25,x]+F[87,x]
F[87,x] = F[85,x]
F[88,x] = 0
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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{26}\! \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_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{14}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{48}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{16}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{43}\! \left(x \right)
F_{32}\! \left(x \right) = 0
F_{33}\! \left(x \right) = F_{16}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{37}\! \left(x \right)
F_{35}\! \left(x \right) = F_{14}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{19}\! \left(x \right)
F_{37}\! \left(x \right) = F_{31}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{16}\! \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_{36}\! \left(x \right)
F_{42}\! \left(x \right) = F_{38}\! \left(x \right)
F_{43}\! \left(x \right) = F_{16}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{14}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{16}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{14}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{32}\! \left(x \right)+F_{49}\! \left(x \right)+F_{79}\! \left(x \right)
F_{49}\! \left(x \right) = F_{16}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{60}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{6}\! \left(x \right)
F_{52}\! \left(x \right) = F_{32}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{53}\! \left(x \right) = F_{16}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{25}\! \left(x \right)+F_{6}\! \left(x \right)
F_{55}\! \left(x \right) = F_{16}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{24}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{16}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{14}\! \left(x \right)
F_{60}\! \left(x \right) = F_{48}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{32}\! \left(x \right)+F_{62}\! \left(x \right)+F_{74}\! \left(x \right)+F_{88}\! \left(x \right)
F_{62}\! \left(x \right) = F_{16}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{68}\! \left(x \right)
F_{64}\! \left(x \right) = F_{52}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{16}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{25}\! \left(x \right)
F_{68}\! \left(x \right) = F_{61}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{16}\! \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_{65}\! \left(x \right)
F_{73}\! \left(x \right) = F_{69}\! \left(x \right)
F_{74}\! \left(x \right) = F_{16}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{87}\! \left(x \right)
F_{76}\! \left(x \right) = F_{32}\! \left(x \right)+F_{77}\! \left(x \right)+F_{79}\! \left(x \right)
F_{77}\! \left(x \right) = F_{16}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{6}\! \left(x \right)+F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = F_{16}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{82}\! \left(x \right)
F_{81}\! \left(x \right) = F_{14}\! \left(x \right)+F_{76}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{84}\! \left(x \right)
F_{83}\! \left(x \right) = F_{46}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{16}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{25}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{85}\! \left(x \right)
F_{88}\! \left(x \right) = 0
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_16(x)*F_4(x))
Eq(F_4(x), F_26(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_16(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_21(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_16(x))
Eq(F_12(x), F_13(x) + F_17(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), x)
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_16(x)*F_20(x))
Eq(F_20(x), F_14(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_16(x)*F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_14(x) + F_25(x))
Eq(F_25(x), F_22(x))
Eq(F_26(x), F_27(x) + F_48(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_16(x)*F_29(x))
Eq(F_29(x), F_13(x) + F_30(x))
Eq(F_30(x), F_27(x) + F_31(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_43(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_16(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_37(x))
Eq(F_35(x), F_14(x) + F_36(x))
Eq(F_36(x), F_19(x))
Eq(F_37(x), F_31(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_16(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_36(x))
Eq(F_42(x), F_38(x))
Eq(F_43(x), F_16(x)*F_44(x))
Eq(F_44(x), F_14(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_16(x)*F_47(x))
Eq(F_47(x), F_14(x) + F_45(x))
Eq(F_48(x), F_32(x) + F_49(x) + F_79(x))
Eq(F_49(x), F_16(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_60(x))
Eq(F_51(x), F_52(x) + F_6(x))
Eq(F_52(x), F_32(x) + F_53(x) + F_55(x))
Eq(F_53(x), F_16(x)*F_54(x))
Eq(F_54(x), F_25(x) + F_6(x))
Eq(F_55(x), F_16(x)*F_56(x))
Eq(F_56(x), F_24(x) + F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_16(x)*F_59(x))
Eq(F_59(x), F_14(x))
Eq(F_60(x), F_48(x) + F_61(x))
Eq(F_61(x), F_32(x) + F_62(x) + F_74(x) + F_88(x))
Eq(F_62(x), F_16(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_68(x))
Eq(F_64(x), F_52(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_16(x)*F_67(x))
Eq(F_67(x), F_25(x))
Eq(F_68(x), F_61(x) + F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_16(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_73(x))
Eq(F_72(x), F_65(x))
Eq(F_73(x), F_69(x))
Eq(F_74(x), F_16(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_87(x))
Eq(F_76(x), F_32(x) + F_77(x) + F_79(x))
Eq(F_77(x), F_16(x)*F_78(x))
Eq(F_78(x), F_6(x) + F_76(x))
Eq(F_79(x), F_16(x)*F_80(x))
Eq(F_80(x), F_81(x) + F_82(x))
Eq(F_81(x), F_14(x) + F_76(x))
Eq(F_82(x), F_83(x) + F_84(x))
Eq(F_83(x), F_46(x))
Eq(F_84(x), F_85(x))
Eq(F_85(x), F_16(x)*F_86(x))
Eq(F_86(x), F_25(x) + F_87(x))
Eq(F_87(x), F_85(x))
Eq(F_88(x), 0)
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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