0123_0213_0231_1302_1320
Counting sequence:
1, 1, 2, 6, 19, 57, 163, 452, 1230, 3308, 8829, 23443, 62021, 163654, 431000, 1133454, 2977567, 7815645, 20502151, 53756360, 140898018, 369199856, 967225857, 2533526311, 6635450249, 17377018762, 45503994668, 119151742482, 311984787235, 816869728113, 2138758614859, 5599674551948, 14660801911926, 38383804925684, 100492760348805, 263098771088059, 688812142849997, 1803354837331150, 4721286728881856, 12360574068791190, 32360572916445223, 84721419558451461, 221804235514723087, 580692386497345616, 1520275123000569354, 3980137380550873592, 10420145814745073673, 27280317655870391887, 71420842337238190865, 186982279724588358418, 489526137574015239764, 1281596414472434071578, 3355263668793240396331, 8784195717807193960089, 22997325736428155169235, 60207785995076898918164, 157626041256001796326302, 412670355787326999542780, 1080385062134776221266061, 2828484902674595702183395, 7405069790004198961140053, 19386724755568377332948566, 50755105053161685341129192, 132878591556838183297286046, 347880671923195873764422959, 910763428824435456423370797, 2384409623773482532360465303, 6242465460942756214367576792, 16342986795948274258161368370, 42786495000689042854954734848, 112016498353692806896379249169, 293263000355537283013535838583, 767772503303214852502933918361, 2010054510734698895212677219994, 5262391031262065074569920348540, 13777118587773862811366729039394, 36068964741504256325269557197107, 94429775655628372095920523406785, 247220362263159791825449174732891, 647231311209408867106341324211100, 1694473571516182536945403444738758, 4436189403641370198633526303681796, 11614094640012390968762490053659797, 30406094517604728527268573032003851, 79604188915219646252272487391764189, 208406472232889913508007405842113502, 545615227793121500828666763532225808, 1428439211165817402091826951549862822, 3739702405743016331674482224707960375, 9790668006140602845386955989755213653, 25632301612833534709397058278920071199, 67106236832669486292625563915729781088, 175686408885793894188122323605718834266, 459952989825950136311026787176325846024, 1204172560594532394823528798473056952345, 3152564691962598808316701129342441507999, 8253521515303167550440857631753460565537, 21607999853966710883634437850316326176290, 56570478046636579181719588087992289938596, 148103434286022254824038590751254087589930, 387739824811588641615424712840957060731963
Generating function in Maple syntax:
(x^5-3*x^4+11*x^3-13*x^2+6*x-1)/(2*x-1)/(x^2-3*x+1)/(x-1)^2
Generating function in latex syntax:
\frac{x^{5}-3 x^{4}+11 x^{3}-13 x^{2}+6 x -1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**5 - 3*x**4 + 11*x**3 - 13*x**2 + 6*x - 1)/((x - 1)**2*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x-1)^2*F(x)-x^5+3*x^4-11*x^3+13*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)-x^{5}+3 x^{4}-11 x^{3}+13 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/2*(5^(1/2)-1)*(3/2-1/2*5^(1/2))^(-n)+1/2*(-5^(1/2)-1)*(3/2+1/2*5^(1/2))^(-n)-n-2^(n-1)+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\sqrt{5}-1\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{2}+\frac{\left(-\sqrt{5}-1\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{2}-n -2^{n -1}+2 & \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) = 57
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)-n+2, 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) = 19
a \! \left(5\right) = 57
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-n +2, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16737/
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[14,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[7,x]
F[14,x] = F[15,x]+F[2,x]
F[15,x] = F[16,x]+F[17,x]+F[85,x]
F[16,x] = 0
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[29,x]
F[19,x] = F[20,x]+F[7,x]
F[20,x] = F[16,x]+F[21,x]+F[27,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]
F[23,x] = F[11,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[11,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[13,x]
F[29,x] = F[15,x]+F[30,x]
F[30,x] = F[16,x]+F[31,x]+F[59,x]+F[84,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[37,x]
F[33,x] = F[20,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[24,x]
F[37,x] = F[30,x]+F[38,x]
F[38,x] = F[16,x]+F[39,x]+F[43,x]+F[57,x]+F[58,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[34,x]
F[42,x] = F[38,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[50,x]
F[48,x] = F[24,x]+F[49,x]
F[49,x] = F[35,x]
F[50,x] = F[45,x]+F[51,x]
F[51,x] = 2*F[16,x]+F[43,x]+F[52,x]+F[56,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[49,x]
F[55,x] = F[51,x]
F[56,x] = 0
F[57,x] = 0
F[58,x] = 0
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[45,x]+F[62,x]
F[62,x] = F[16,x]+F[63,x]+F[82,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[68,x]
F[65,x] = F[11,x]+F[66,x]
F[66,x] = F[16,x]+F[21,x]+F[67,x]
F[67,x] = F[13,x]*F[4,x]
F[68,x] = F[62,x]+F[69,x]
F[69,x] = F[16,x]+F[59,x]+F[70,x]+F[81,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[72,x]+F[74,x]
F[72,x] = F[66,x]+F[73,x]
F[73,x] = F[35,x]
F[74,x] = F[69,x]+F[75,x]
F[75,x] = F[16,x]+F[43,x]+F[57,x]+F[76,x]+F[80,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[78,x]+F[79,x]
F[78,x] = F[73,x]
F[79,x] = F[75,x]
F[80,x] = 0
F[81,x] = 0
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[2,x]
F[84,x] = 0
F[85,x] = F[4,x]*F[86,x]
F[86,x] = F[87,x]
F[87,x] = F[2,x]+F[62,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_{14}\! \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_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{85}\! \left(x \right)
F_{16}\! \left(x \right) = 0
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_{29}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{7}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{27}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{16}\! \left(x \right)+F_{31}\! \left(x \right)+F_{59}\! \left(x \right)+F_{84}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{33}\! \left(x \right) = F_{20}\! \left(x \right)+F_{34}\! \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_{24}\! \left(x \right)
F_{37}\! \left(x \right) = F_{30}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{16}\! \left(x \right)+F_{39}\! \left(x \right)+F_{43}\! \left(x \right)+F_{57}\! \left(x \right)+F_{58}\! \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_{34}\! \left(x \right)
F_{42}\! \left(x \right) = F_{38}\! \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_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{50}\! \left(x \right)
F_{48}\! \left(x \right) = F_{24}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{35}\! \left(x \right)
F_{50}\! \left(x \right) = F_{45}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{43}\! \left(x \right)+F_{52}\! \left(x \right)+F_{56}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{49}\! \left(x \right)
F_{55}\! \left(x \right) = F_{51}\! \left(x \right)
F_{56}\! \left(x \right) = 0
F_{57}\! \left(x \right) = 0
F_{58}\! \left(x \right) = 0
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{45}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{16}\! \left(x \right)+F_{63}\! \left(x \right)+F_{82}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{65}\! \left(x \right) = F_{11}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{68}\! \left(x \right) = F_{62}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{16}\! \left(x \right)+F_{59}\! \left(x \right)+F_{70}\! \left(x \right)+F_{81}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{74}\! \left(x \right)
F_{72}\! \left(x \right) = F_{66}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{35}\! \left(x \right)
F_{74}\! \left(x \right) = F_{69}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{16}\! \left(x \right)+F_{43}\! \left(x \right)+F_{57}\! \left(x \right)+F_{76}\! \left(x \right)+F_{80}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \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_{73}\! \left(x \right)
F_{79}\! \left(x \right) = F_{75}\! \left(x \right)
F_{80}\! \left(x \right) = 0
F_{81}\! \left(x \right) = 0
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{2}\! \left(x \right)
F_{84}\! \left(x \right) = 0
F_{85}\! \left(x \right) = F_{4}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{2}\! \left(x \right)+F_{62}\! \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_14(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_7(x))
Eq(F_14(x), F_15(x) + F_2(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_85(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_29(x))
Eq(F_19(x), F_20(x) + F_7(x))
Eq(F_20(x), F_16(x) + F_21(x) + F_27(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_11(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_11(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_13(x))
Eq(F_29(x), F_15(x) + F_30(x))
Eq(F_30(x), F_16(x) + F_31(x) + F_59(x) + F_84(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_37(x))
Eq(F_33(x), F_20(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_24(x))
Eq(F_37(x), F_30(x) + F_38(x))
Eq(F_38(x), F_16(x) + F_39(x) + F_43(x) + F_57(x) + F_58(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_34(x))
Eq(F_42(x), F_38(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_50(x))
Eq(F_48(x), F_24(x) + F_49(x))
Eq(F_49(x), F_35(x))
Eq(F_50(x), F_45(x) + F_51(x))
Eq(F_51(x), 2*F_16(x) + F_43(x) + F_52(x) + F_56(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_49(x))
Eq(F_55(x), F_51(x))
Eq(F_56(x), 0)
Eq(F_57(x), 0)
Eq(F_58(x), 0)
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_45(x) + F_62(x))
Eq(F_62(x), F_16(x) + F_63(x) + F_82(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_68(x))
Eq(F_65(x), F_11(x) + F_66(x))
Eq(F_66(x), F_16(x) + F_21(x) + F_67(x))
Eq(F_67(x), F_13(x)*F_4(x))
Eq(F_68(x), F_62(x) + F_69(x))
Eq(F_69(x), F_16(x) + F_59(x) + F_70(x) + F_81(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_74(x))
Eq(F_72(x), F_66(x) + F_73(x))
Eq(F_73(x), F_35(x))
Eq(F_74(x), F_69(x) + F_75(x))
Eq(F_75(x), F_16(x) + F_43(x) + F_57(x) + F_76(x) + F_80(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_78(x) + F_79(x))
Eq(F_78(x), F_73(x))
Eq(F_79(x), F_75(x))
Eq(F_80(x), 0)
Eq(F_81(x), 0)
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_2(x))
Eq(F_84(x), 0)
Eq(F_85(x), F_4(x)*F_86(x))
Eq(F_86(x), F_87(x))
Eq(F_87(x), F_2(x) + F_62(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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