0132_0231_1032_1203_3120
Counting sequence:
1, 1, 2, 6, 19, 53, 134, 322, 750, 1710, 3838, 8510, 18686, 40702, 88062, 189438, 405502, 864254, 1835006, 3883006, 8191998, 17235966, 36175870, 75759614, 158334974, 330301438, 687865854, 1430257662, 2969567230, 6157238270, 12750684158, 26373783550, 54492397566, 112474456062, 231928233982, 477815111678, 983547510782, 2022929596414, 4157528342526, 8538394984446, 17523466567678, 35940286332926, 73667279060990, 150907970912254, 308962767405054, 632219185971198, 1293025674264574, 2643225953173502, 5400801115635710, 11030300649848830, 22517998136852478, 45950789948014590, 93731167244648446, 191121509186535422, 389561367767547902, 793759434324049918, 1616792266226008062, 3292131327607832574, 6701356245527298046, 13636899671677861886, 27742173704602255358, 56421096131697573886, 114715689708381274110, 233178374306734800894, 473850738393414107134, 962689456346717224958, 1955354871813212471294, 3970661661865980985342, 8061227160211074056190, 16362261993380372283390, 33204139332677192908798, 67367509357187282501630, 136653480098040358371326, 277143882963412303478782, 561961611461487780429822, 1139270913992301907804158, 2309237210123256509497342, 4679865184523818406772734, 9482511897602247589101566, 19210586852313716729315326, 38912299818845876560855038, 78806851866128639326158846, 159578208189131051061215230, 323085425292009646940225534, 654028868411514383516041214, 1323773772478018946303262718, 2678979616266018251148886014, 5420823375151997219382493182, 10967375035543915872934428670, 22186206641567674614207741950, 44875326424095034965093253118, 90756479130109441403542044670, 183524610824057625753795166206, 371072526775792737401012486142, 750191663806940446588869279742, 1516476548124590836751427174398, 3065139537270601560650231578622, 6194651956584042895595217616894, 12518049677253765339779944153086, 25293590882678889776738906144766, 51102164821700497747835847966718
Generating function in Maple syntax:
(3*x^6+2*x^5-x^4-5*x^2+4*x-1)/(x-1)/(2*x-1)^2
Generating function in latex syntax:
\frac{3 x^{6}+2 x^{5}-x^{4}-5 x^{2}+4 x -1}{\left(x -1\right) \left(2 x -1\right)^{2}}
Generating function in sympy syntax:
(3*x**6 + 2*x**5 - x**4 - 5*x**2 + 4*x - 1)/((x - 1)*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(x-1)*(2*x-1)^2*F(x)+3*x^6+2*x^5-x^4-5*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+3 x^{6}+2 x^{5}-x^{4}-5 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,-2+1/32*(13*n-10)*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ -2+\frac{\left(13 n -10\right) 2^{n}}{32} & \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) = 53
a(6) = 134
a(n+2) = -4*a(n)+4*a(n+1)-2, n >= 7
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) = 53
a \! \left(6\right) = 134
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)-2, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18859/
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[22,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[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[17,x]
F[21,x] = F[12,x]*F[13,x]
F[22,x] = F[23,x]+F[40,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[34,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[9,x]
F[30,x] = F[14,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[10,x]+F[31,x]
F[34,x] = F[23,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[10,x]
F[39,x] = F[36,x]
F[40,x] = F[18,x]+F[41,x]+F[61,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[43,x]+F[48,x]
F[43,x] = F[10,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[14,x]+F[47,x]
F[47,x] = F[45,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[18,x]+F[41,x]+F[50,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[23,x]+F[49,x]
F[52,x] = F[53,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[55,x]+F[60,x]
F[55,x] = F[56,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[12,x]
F[59,x] = F[56,x]
F[60,x] = F[53,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[51,x]+F[63,x]
F[63,x] = F[64,x]+F[73,x]
F[64,x] = F[18,x]+F[65,x]+F[71,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[67,x]+F[70,x]
F[67,x] = F[12,x]+F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[12,x]*F[14,x]
F[70,x] = F[56,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[23,x]+F[64,x]
F[73,x] = F[18,x]+F[74,x]+F[75,x]+F[77,x]
F[74,x] = 0
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[49,x]+F[73,x]
F[77,x] = F[12,x]*F[63,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_{22}\! \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_{17}\! \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_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{40}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{34}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{9}\! \left(x \right)
F_{30}\! \left(x \right) = F_{14}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{23}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right)
F_{39}\! \left(x \right) = F_{36}\! \left(x \right)
F_{40}\! \left(x \right) = F_{18}\! \left(x \right)+F_{41}\! \left(x \right)+F_{61}\! \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_{48}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \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_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{18}\! \left(x \right)+F_{41}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{23}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \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_{60}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right)
F_{59}\! \left(x \right) = F_{56}\! \left(x \right)
F_{60}\! \left(x \right) = F_{53}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{51}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{73}\! \left(x \right)
F_{64}\! \left(x \right) = F_{18}\! \left(x \right)+F_{65}\! \left(x \right)+F_{71}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{70}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{70}\! \left(x \right) = F_{56}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{23}\! \left(x \right)+F_{64}\! \left(x \right)
F_{73}\! \left(x \right) = F_{18}\! \left(x \right)+F_{74}\! \left(x \right)+F_{75}\! \left(x \right)+F_{77}\! \left(x \right)
F_{74}\! \left(x \right) = 0
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{49}\! \left(x \right)+F_{73}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \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_22(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_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_17(x))
Eq(F_21(x), F_12(x)*F_13(x))
Eq(F_22(x), F_23(x) + F_40(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_34(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_9(x))
Eq(F_30(x), F_14(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_10(x) + F_31(x))
Eq(F_34(x), F_23(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_10(x))
Eq(F_39(x), F_36(x))
Eq(F_40(x), F_18(x) + F_41(x) + F_61(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_48(x))
Eq(F_43(x), F_10(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_14(x) + F_47(x))
Eq(F_47(x), F_45(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_18(x) + F_41(x) + F_50(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_23(x) + F_49(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_60(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_12(x))
Eq(F_59(x), F_56(x))
Eq(F_60(x), F_53(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_51(x) + F_63(x))
Eq(F_63(x), F_64(x) + F_73(x))
Eq(F_64(x), F_18(x) + F_65(x) + F_71(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_70(x))
Eq(F_67(x), F_12(x) + F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_12(x)*F_14(x))
Eq(F_70(x), F_56(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_23(x) + F_64(x))
Eq(F_73(x), F_18(x) + F_74(x) + F_75(x) + F_77(x))
Eq(F_74(x), 0)
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_49(x) + F_73(x))
Eq(F_77(x), F_12(x)*F_63(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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