0132_0213_0231_1203_3021
Counting sequence:
1, 1, 2, 6, 19, 58, 170, 483, 1342, 3670, 9923, 26610, 70930, 188227, 497846, 1313502, 3459043, 9096394, 23895674, 62721699, 164531566, 431397286, 1130708867, 2962826466, 7761964834, 20331456643, 53249182310, 139449644718, 365166860707, 956185155130, 2503657040138, 6555322836195, 17163385210270, 44936980278262, 117651850591811, 308027161431762, 806446813572658, 2111347639024579, 5527664822977814, 14471784268862334, 37887962861516131, 99192654071499946, 259691098864611482, 679882841545590051, 1779961823818669774, 4660011426003441478, 12200090046377699075, 31940293897501744578, 83620862014871712322, 218922432884601747715, 573146718113910241478, 1500518284407082398030, 3928409261007243795235, 10284711750414462672922, 26925730493835771594026, 70492488738292106850147, 184551753735439058438398, 483162808496822087429014, 1264936743812621241776579, 3311647567056229713756594, 8670006245586444051204946, 22698371746163854743281731, 59425110145826624785487222, 155576960997159028826873886, 407305777457336480122522339, 1066340380598222448395468938, 2791715382784074938773436090, 7308805804647490515343942563, 19134702104945372902096598062, 50095300657762580780622264550, 131151200163490274619123021443, 343358300423004053435452451490, 898923702286113506404645636450, 2353412808796519707213307064707, 6161314728825812098104920771366, 16130531387125649552840745676782, 42230279451440602491895897113763, 110560307004975089785804107374074, 289450641639042530591430748427594, 757791618063268229440316784746979, 1983924212852993612633176899489886, 5193981021100175518266528501075766, 13598018851656458761781037778443587, 35600075536287052406305843183667346, 93202207762040401735595008471383154, 244006547759505559357396215628131523, 638817435535819089450427705208310230, 1672445758886637335221555033587396798, 4378519841201464168669573662735075427, 11463113764872497675697838488980220010, 30010821453725513868245286872930365658, 78569350596923013948680712267260439075, 205697230338281468017082230203750075790, 538522340420397270181136738893788036742, 1409869790927862102683469507577210531331, 3691087032373092558183554826037036551042, 9663391306211222612495761054932285109378, 25299086886300189360560860507556590752259, 66233869352768573631701084805331031097734, 173402521172163987859570922583623590441614, 453973694164040302597068740295913916028451
Generating function in Maple syntax:
-(3*x^3-8*x^2+5*x-1)/(-1+x)/(2*x-1)/(x^2-3*x+1)
Generating function in latex syntax:
-\frac{3 x^{3}-8 x^{2}+5 x -1}{\left(-1+x \right) \left(2 x -1\right) \left(x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(-3*x**3 + 8*x**2 - 5*x + 1)/((x - 1)*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(-1+x)*(2*x-1)*(x^2-3*x+1)*F(x)+3*x^3-8*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(-1+x \right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+3 x^{3}-8 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
15*(3/2-1/2*5^(1/2))^(-n)*5^(1/2)-15*(3/2+1/2*5^(1/2))^(-n)*5^(1/2)-2^n+149/5*(3/2-1/2*5^(1/2))^(-n+1)+149/5*(3/2+1/2*5^(1/2))^(-n+1)-221/5*(3/2-1/2*5^(1/2))^(-n)-221/5*(3/2+1/2*5^(1/2))^(-n)+1
Explicit closed form in latex syntax:
15 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}-15 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}-2^{n}+\frac{149 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n +1}}{5}+\frac{149 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n +1}}{5}-\frac{221 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{5}-\frac{221 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{5}+1
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)+1, 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 +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+1, \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18160/
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[19,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[18,x] = F[12,x]*F[13,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[48,x]
F[21,x] = 0
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[28,x]
F[24,x] = F[25,x]+F[6,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[13,x]
F[28,x] = F[20,x]+F[29,x]
F[29,x] = 2*F[21,x]+F[30,x]+F[34,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[25,x]
F[33,x] = F[29,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]+F[42,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[40,x]+F[41,x]
F[40,x] = F[14,x]
F[41,x] = F[37,x]
F[42,x] = 2*F[21,x]+F[43,x]+F[47,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[17,x]
F[46,x] = F[42,x]
F[47,x] = F[12,x]*F[36,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[36,x]+F[50,x]
F[50,x] = F[2,x]+F[51,x]
F[51,x] = F[21,x]+F[52,x]+F[62,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]+F[56,x]
F[54,x] = F[10,x]+F[55,x]
F[55,x] = F[18,x]
F[56,x] = F[51,x]+F[57,x]
F[57,x] = 2*F[21,x]+F[47,x]+F[58,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[55,x]
F[61,x] = F[57,x]
F[62,x] = F[12,x]*F[50,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_{19}\! \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_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{48}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{6}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)
F_{28}\! \left(x \right) = F_{20}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{25}\! \left(x \right)
F_{33}\! \left(x \right) = F_{29}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \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_{42}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{14}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right)
F_{42}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{43}\! \left(x \right)+F_{47}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{17}\! \left(x \right)
F_{46}\! \left(x \right) = F_{42}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{36}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{2}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{21}\! \left(x \right)+F_{52}\! \left(x \right)+F_{62}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{56}\! \left(x \right)
F_{54}\! \left(x \right) = F_{10}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{18}\! \left(x \right)
F_{56}\! \left(x \right) = F_{51}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{47}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{55}\! \left(x \right)
F_{61}\! \left(x \right) = F_{57}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \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_19(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))
Eq(F_18(x), F_12(x)*F_13(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_48(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_28(x))
Eq(F_24(x), F_25(x) + F_6(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_13(x))
Eq(F_28(x), F_20(x) + F_29(x))
Eq(F_29(x), 2*F_21(x) + F_30(x) + F_34(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_25(x))
Eq(F_33(x), F_29(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x) + F_42(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_41(x))
Eq(F_40(x), F_14(x))
Eq(F_41(x), F_37(x))
Eq(F_42(x), 2*F_21(x) + F_43(x) + F_47(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_17(x))
Eq(F_46(x), F_42(x))
Eq(F_47(x), F_12(x)*F_36(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_36(x) + F_50(x))
Eq(F_50(x), F_2(x) + F_51(x))
Eq(F_51(x), F_21(x) + F_52(x) + F_62(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_56(x))
Eq(F_54(x), F_10(x) + F_55(x))
Eq(F_55(x), F_18(x))
Eq(F_56(x), F_51(x) + F_57(x))
Eq(F_57(x), 2*F_21(x) + F_47(x) + F_58(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_55(x))
Eq(F_61(x), F_57(x))
Eq(F_62(x), F_12(x)*F_50(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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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