0123_0132_1032_1203_1302
Counting sequence:
1, 1, 2, 6, 19, 57, 166, 480, 1389, 4025, 11670, 33838, 98111, 284457, 824730, 2391152, 6932713, 20100161, 58276826, 168963238, 489878683, 1420315593, 4117950942, 11939262000, 34615754101, 100362185881, 290982202062, 843650834926, 2446014657319, 7091781879577, 20561352760946, 59613963675040, 172840022072849, 501118720992705, 1452904075790450, 4212435427011718, 12213202868943331, 35409996640300057, 102664950015240790, 297658654664661312, 863008014747225533, 2502137337001038521, 7254499548359815366, 21033123529594219758, 60981778613686920143, 176805756770091470793, 512616659233166457482, 1486240290609268006800, 4309087817658276059001, 12493429183432456594561, 36222462703548424613322, 105020550006389692199846, 304488295395888259594219, 882809336148523575398697, 2559547725726869072230926, 7420950699111548198717200, 21515718861231651916336709, 62380977436082676590338585, 180862483423354064165672830, 524378412370660402332727598, 1520341389522622357765232951, 4407957852890638626651705657, 12780085161636739944377018850, 37053570426400310388764446272, 107430198154358181388618353953, 311474639087995602514821374081, 903064989748987045226717991906, 2618275369379082444013587625478, 7591220994850730981565739636403, 22009387120472563843176865401593, 63812280230994483379698396303622, 185012289801168222553231190506400, 536410033516490815362050647250701, 1555224922443751901837851484986041, 4509096415542375274389364128257782, 13073318329228643343483383058094766, 37903747533149066895413829336570719, 109895134569210196658338364855405417, 318621281218244181013054272319786106, 923785399991662714553174671588758960, 2678350491765243594372572156123272969, 7765398064100022077962036847633033665, 22514382370540683544477262381536346810, 65276423609284795769359221119448557478, 189257311574941736649824480803038182971, 548717714667809152688777600345064229257, 1590908841960579939239008182788537646910, 4612555555930251415269930512499418594032, 13373279596789245946151069000336690429973, 38773431562024947154691347917271226866841, 112416627814763887603939984875525520582446, 325931899760464505104566616346189974316398, 944981230503641610258040290717536959099463, 2739804010164261142976466015153710207163993, 7943571545978171028686776115312598461473362, 23030964504023387268461887598416816909752800, 66774161083013028855311418253441831166545201, 193599733418079169547748140865878964157280321, 561307789894289508335357441761894764243919442, 1627411512574890800129521927689363679589399686, 4718388518641576535127465643695075420863026819
Generating function in Maple syntax:
-(2*x-1)*(x-1)/(x^4+2*x^3-4*x^2+4*x-1)
Generating function in latex syntax:
-\frac{\left(2 x -1\right) \left(x -1\right)}{x^{4}+2 x^{3}-4 x^{2}+4 x -1}
Generating function in sympy syntax:
(1 - 2*x)*(x - 1)/(x**4 + 2*x**3 - 4*x**2 + 4*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^4+2*x^3-4*x^2+4*x-1)*F(x)+(2*x-1)*(x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{4}+2 x^{3}-4 x^{2}+4 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right) = 0
Explicit closed form in Maple syntax:
-13/326*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^4+2*_Z^3-4*_Z^2+4*_Z-1))-29/326*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^4+2*_Z^3-4*_Z^2+4*_Z-1))+54/163*Sum(_alpha^(-n),_alpha = RootOf(_Z^4+2*_Z^3-4*_Z^2+4*_Z-1))-1/163*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^4+2*_Z^3-4*_Z^2+4*_Z-1))
Explicit closed form in latex syntax:
-\frac{13 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+2 \textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{326}-\frac{29 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+2 \textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{326}+\frac{54 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+2 \textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{163}-\frac{\left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+2 \textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{163}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = a(n)+2*a(n+1)-4*a(n+2)+4*a(n+3), n >= 4
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(n +4\right) = a \! \left(n \right)+2 a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16334/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[12,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[55,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[30,x]
F[23,x] = F[24,x]+F[26,x]
F[24,x] = F[25,x]
F[25,x] = F[10,x]*F[4,x]
F[26,x] = F[13,x]+F[14,x]+F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[7,x]
F[29,x] = F[18,x]
F[30,x] = F[31,x]+F[36,x]
F[31,x] = F[13,x]+F[21,x]+F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[2,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[2,x]*F[4,x]
F[36,x] = F[13,x]+F[37,x]+F[59,x]+F[62,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[45,x]
F[39,x] = F[40,x]+F[42,x]
F[40,x] = F[41,x]
F[41,x] = F[16,x]*F[4,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[29,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = 2*F[13,x]+F[37,x]+F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[34,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[33,x]+F[57,x]
F[57,x] = F[54,x]+F[58,x]
F[58,x] = 2*F[13,x]+F[53,x]+F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[48,x]+F[61,x]
F[61,x] = F[58,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[52,x]+F[54,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{55}\! \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_{30}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{36}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{13}\! \left(x \right)+F_{37}\! \left(x \right)+F_{59}\! \left(x \right)+F_{62}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{45}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{29}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{37}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{34}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{33}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{54}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{53}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{48}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{58}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{52}\! \left(x \right)+F_{54}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_12(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_55(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_30(x))
Eq(F_23(x), F_24(x) + F_26(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_10(x)*F_4(x))
Eq(F_26(x), F_13(x) + F_14(x) + F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_7(x))
Eq(F_29(x), F_18(x))
Eq(F_30(x), F_31(x) + F_36(x))
Eq(F_31(x), F_13(x) + F_21(x) + F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_2(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_2(x)*F_4(x))
Eq(F_36(x), F_13(x) + F_37(x) + F_59(x) + F_62(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_45(x))
Eq(F_39(x), F_40(x) + F_42(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_16(x)*F_4(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_29(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), 2*F_13(x) + F_37(x) + F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_34(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_33(x) + F_57(x))
Eq(F_57(x), F_54(x) + F_58(x))
Eq(F_58(x), 2*F_13(x) + F_53(x) + F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_48(x) + F_61(x))
Eq(F_61(x), F_58(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_52(x) + F_54(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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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