01234_01243_01324_01342_01423_01432_02341_02431_10234_10243_10324_10342_10423_10432_12340_12430_20134_20143_20314_20341_20413_20431_21340_21430_30124_30142_30214_30241_30412_30421_31240_31420_40123_40132_40213_40231_40312_40321_41230_41320
Counting sequence:
1, 1, 2, 6, 24, 80, 240, 672, 1792, 4608, 11520, 28160, 67584, 159744, 372736, 860160, 1966080, 4456448, 10027008, 22413312, 49807360, 110100480, 242221056, 530579456, 1157627904, 2516582400, 5452595200, 11777605632, 25367150592, 54492397568, 116769423360, 249644974080, 532575944704, 1133871366144, 2409476653056, 5111011082240, 10823317585920, 22883585753088, 48309792145408, 101842264522752, 214404767416320, 450799767388160, 946679511515136, 1985717999763456, 4160551999504384, 8708132091985920, 18207912555970560, 38034306228027392, 79375943432404992, 165507286305865728, 344806846470553600, 717761190612172800, 1492943276473319424, 3102980143258271744, 6444651066767179776, 13375690893290373120, 27742173704602255360, 57501960042266492928, 119111202944694878208, 246581086797789396992, 510167765788529786880, 1054923176715264983040, 2180174565211547631616, 4503311396994344288256, 9297159013149614014464, 19184613836657933680640, 39568266038106988216320, 81571502293943637245952, 168086731999641434324992, 346208492775380864729088, 712782191008137074442240, 1466885088741383544504320, 3017592182553703291551744, 6205189558490713810796544, 12755111870230911722192896, 26209133979926530936012800, 53834977904713955436134400, 110541154631012655162195968, 226900264768920713227665408, 465587556279084060908716032, 955051397495557048017879040, 1958459827775699263024005120, 4014842646940183489199210496, 8227949128297166163050233856, 16857261628706389212102918144, 34526921408193809229608386560, 70697981931063514136817172480, 144722927717706487762425741312, 296177154399027230769615470592, 605971649230193644563121307648, 1239487464334487000242748129280, 2534682230436816112855956848640, 5182017004448601830727734001664, 10591814976125713632036906860544, 21644143646865588726336287932416, 44219218203273783419396717281280, 90320105266261344856640103383040, 184443162333207377917770316382208, 376571456430298396582114395947008, 768671632713392603332563406159872, 1568717617782433884352170216652800
Generating function in Maple syntax:
(8*x^5-4*x^4+2*x^3-8*x^2+5*x-1)/(2*x-1)^3
Generating function in latex syntax:
\frac{8 x^{5}-4 x^{4}+2 x^{3}-8 x^{2}+5 x -1}{\left(2 x -1\right)^{3}}
Generating function in sympy syntax:
(8*x**5 - 4*x**4 + 2*x**3 - 8*x**2 + 5*x - 1)/(2*x - 1)**3
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^3*F(x)+8*x^5-4*x^4+2*x^3-8*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{3} F \! \left(x \right)+8 x^{5}-4 x^{4}+2 x^{3}-8 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/8*2^n*n*(-1+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{2^{n} n \left(-1+n \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) = 24
a(5) = 80
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2), n >= 6
Recurrence in latex format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 80
a{\left(n + 3 \right)} = 8 a{\left(n \right)} - 12 a{\left(n + 1 \right)} + 6 a{\left(n + 2 \right)}, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/25591/
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[52,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[16,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[14,x]+F[15,x]
F[14,x] = F[1,x]+F[4,x]
F[15,x] = F[4,x]
F[16,x] = F[17,x]+F[7,x]
F[17,x] = F[18,x]+F[19,x]+F[49,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[27,x]
F[21,x] = F[11,x]+F[22,x]
F[22,x] = F[18,x]+F[23,x]+F[25,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[4,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[15,x]
F[27,x] = F[17,x]+F[28,x]
F[28,x] = F[18,x]+F[29,x]+F[33,x]+F[46,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[22,x]
F[32,x] = F[28,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[40,x]
F[38,x] = F[39,x]+F[4,x]
F[39,x] = F[23,x]
F[40,x] = F[35,x]+F[41,x]
F[41,x] = 2*F[18,x]+F[33,x]+F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[39,x]
F[45,x] = F[41,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[35,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[48,x]+F[51,x]
F[51,x] = F[35,x]+F[7,x]
F[52,x] = F[53,x]+F[7,x]
F[53,x] = F[18,x]+F[19,x]+F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[16,x]+F[56,x]
F[56,x] = F[53,x]+F[57,x]
F[57,x] = F[18,x]+F[29,x]+F[58,x]+F[80,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[27,x]+F[60,x]
F[60,x] = F[57,x]+F[61,x]
F[61,x] = F[18,x]+F[62,x]+F[63,x]+F[66,x]+F[77,x]
F[62,x] = 0
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[32,x]+F[65,x]
F[65,x] = F[61,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[68,x]
F[68,x] = 2*F[18,x]+F[42,x]+F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[40,x]+F[71,x]
F[71,x] = F[68,x]+F[72,x]
F[72,x] = 2*F[18,x]+F[66,x]+F[73,x]+F[74,x]
F[73,x] = 0
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[45,x]+F[76,x]
F[76,x] = F[72,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[79,x]
F[79,x] = F[68,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[79,x]+F[82,x]
F[82,x] = F[53,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{52}\! \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_{16}\! \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_{14}\! \left(x \right)+F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{49}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{27}\! \left(x \right)
F_{21}\! \left(x \right) = F_{11}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right)
F_{27}\! \left(x \right) = F_{17}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{18}\! \left(x \right)+F_{29}\! \left(x \right)+F_{33}\! \left(x \right)+F_{46}\! \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_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{22}\! \left(x \right)
F_{32}\! \left(x \right) = F_{28}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{40}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{23}\! \left(x \right)
F_{40}\! \left(x \right) = F_{35}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{39}\! \left(x \right)
F_{45}\! \left(x \right) = F_{41}\! \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_{48}\! \left(x \right) = F_{35}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{35}\! \left(x \right)+F_{7}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{7}\! \left(x \right)
F_{53}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{16}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{53}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{18}\! \left(x \right)+F_{29}\! \left(x \right)+F_{58}\! \left(x \right)+F_{80}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{27}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{57}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{18}\! \left(x \right)+F_{62}\! \left(x \right)+F_{63}\! \left(x \right)+F_{66}\! \left(x \right)+F_{77}\! \left(x \right)
F_{62}\! \left(x \right) = 0
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{32}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{61}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{40}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{68}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{66}\! \left(x \right)+F_{73}\! \left(x \right)+F_{74}\! \left(x \right)
F_{73}\! \left(x \right) = 0
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{45}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{72}\! \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_{79}\! \left(x \right) = F_{68}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{79}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{53}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_52(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_16(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_14(x) + F_15(x))
Eq(F_14(x), F_1(x) + F_4(x))
Eq(F_15(x), F_4(x))
Eq(F_16(x), F_17(x) + F_7(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_49(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_27(x))
Eq(F_21(x), F_11(x) + F_22(x))
Eq(F_22(x), F_18(x) + F_23(x) + F_25(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_4(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_15(x))
Eq(F_27(x), F_17(x) + F_28(x))
Eq(F_28(x), F_18(x) + F_29(x) + F_33(x) + F_46(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_22(x))
Eq(F_32(x), F_28(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_40(x))
Eq(F_38(x), F_39(x) + F_4(x))
Eq(F_39(x), F_23(x))
Eq(F_40(x), F_35(x) + F_41(x))
Eq(F_41(x), 2*F_18(x) + F_33(x) + F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_39(x))
Eq(F_45(x), F_41(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_35(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_48(x) + F_51(x))
Eq(F_51(x), F_35(x) + F_7(x))
Eq(F_52(x), F_53(x) + F_7(x))
Eq(F_53(x), F_18(x) + F_19(x) + F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_16(x) + F_56(x))
Eq(F_56(x), F_53(x) + F_57(x))
Eq(F_57(x), F_18(x) + F_29(x) + F_58(x) + F_80(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_27(x) + F_60(x))
Eq(F_60(x), F_57(x) + F_61(x))
Eq(F_61(x), F_18(x) + F_62(x) + F_63(x) + F_66(x) + F_77(x))
Eq(F_62(x), 0)
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_32(x) + F_65(x))
Eq(F_65(x), F_61(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), 2*F_18(x) + F_42(x) + F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_40(x) + F_71(x))
Eq(F_71(x), F_68(x) + F_72(x))
Eq(F_72(x), 2*F_18(x) + F_66(x) + F_73(x) + F_74(x))
Eq(F_73(x), 0)
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_45(x) + F_76(x))
Eq(F_76(x), F_72(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_68(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_79(x) + F_82(x))
Eq(F_82(x), F_53(x) + F_68(x))
Pack JSON:
{"name": "point_placements", "initial_strats": [{"class_module": "tilings.strategies.factor", "strategy_class": "FactorFactory", "interleaving": null, "unions": false, "ignore_parent": true, "workable": true, "tracked": false}, {"class_module": "tilings.strategies.requirement_insertion", "strategy_class": "PointCorroborationFactory", "ignore_parent": true}], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "strategy_class": "RowColumnSeparationStrategy", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "strategy_class": "InsertionEncodingVerificationStrategy", "ignore_parent": false}, {"class_module": "tilings.strategies.verification", "strategy_class": "OneByOneVerificationStrategy", "ignore_parent": false, "basis": [], "symmetry": false}, {"class_module": "tilings.strategies.verification", "strategy_class": "LocallyFactorableVerificationStrategy", "ignore_parent": false, "basis": [], "symmetry": false}], "expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "strategy_class": "CellInsertionFactory", "ignore_parent": false, "maxreqlen": 1, "extra_basis": [], "one_cell_only": false}, {"class_module": "tilings.strategies.requirement_placement", "strategy_class": "PatternPlacementFactory", "partial": false, "ignore_parent": false, "dirs": [0, 1, 2, 3], "point_only": false}]], "symmetries": [], "iterative": false}
Specification JSON:
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