0132_0321_1032_3120
Counting sequence:
1, 1, 2, 6, 20, 63, 182, 488, 1237, 3010, 7109, 16421, 37290, 83549, 185160, 406646, 886259, 1918924, 4131343, 8850635, 18878480, 40113135, 84940922, 179314116, 377496545, 792734438, 1660957417, 3472899073, 7247775302, 15099515345, 31406972604, 65229843746, 135291501855, 280246652624, 579820626515, 1198295922631, 2473901215564, 5102421207299, 10514080007438, 21646635246480, 44530221008045, 91534343104650, 188016488451917, 385928581462301, 791648372123170, 1622879162734213, 3324923162544752, 6808175999353358, 13933011347357067, 28499341392149780, 58265320179318999, 119063915148839171, 243194379878258120, 496521858917869399, 1013309916158655970, 2067152228963375516, 4215369251219126969, 8592868089023275374, 17509995351216885233, 35668509048774754425, 72634054790231816510, 147862182965828614265, 300912512702387584612, 612201318946236304186, 1245155224975395331895, 2531815624116636596888, 5146641596564965580187, 10459303889793316489343, 21250649172913404230532, 43165381132480351598315, 87658927838267790146294, 177974186823149754910696, 361261035939527859822085, 733147396465512420457874, 1487545442103938243405525, 3017592182553703292705301, 6120186961799060198168474, 12410379116981427622879149, 25160768620729469699928728, 51001558014992168309346470, 103363157577050794438883875, 209446398248234504519429980, 424332962684734840323535007, 859546257746001343217843771, 1740853180245066011578734720, 3525227689996258673445142463, 7137498039004770647467291722, 14449081396034047896090343028, 29246333428117108994494039697, 59189008128332244393616712950, 119770698800860541596492714489, 242326762690113188811506126321, 490224255557010588860055869750, 991589971467589600194201302049, 2005462863642316045336584167180, 4055491568698905780569534011986, 8200114820226358940931802048079, 16578493006109812641449074934624, 33513512743533814802069094461923, 67740078949696008642480081154615, 136906264824648775361643949950140
Generating function in Maple syntax:
-(x^2-x+1)*(x^6+x^5+8*x^4-22*x^3+19*x^2-7*x+1)/(2*x-1)^2/(x-1)^5
Generating function in latex syntax:
-\frac{\left(x^{2}-x +1\right) \left(x^{6}+x^{5}+8 x^{4}-22 x^{3}+19 x^{2}-7 x +1\right)}{\left(2 x -1\right)^{2} \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-x**2 + x - 1)*(x**6 + x**5 + 8*x**4 - 22*x**3 + 19*x**2 - 7*x + 1)/((x - 1)**5*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^5*F(x)+(x^2-x+1)*(x^6+x^5+8*x^4-22*x^3+19*x^2-7*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)+\left(x^{2}-x +1\right) \left(x^{6}+x^{5}+8 x^{4}-22 x^{3}+19 x^{2}-7 x +1\right) = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,9/8*2^n*n-9*2^(-1+n)+47/24*n^2-7/12*n+7-5/12*n^3+1/24*n^4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{9 \,2^{n} n}{8}-9 \,2^{-1+n}+\frac{47 n^{2}}{24}-\frac{7 n}{12}+7-\frac{5 n^{3}}{12}+\frac{n^{4}}{24} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 63
a(6) = 182
a(7) = 488
a(8) = 1237
a(n+2) = 1/24*n^4-3/4*n^3+107/24*n^2-4*a(n)+4*a(n+1)-31/4*n+7, n >= 9
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) = 20
a \! \left(5\right) = 63
a \! \left(6\right) = 182
a \! \left(7\right) = 488
a \! \left(8\right) = 1237
a \! \left(n +2\right) = \frac{n^{4}}{24}-\frac{3 n^{3}}{4}+\frac{107 n^{2}}{24}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{31 n}{4}+7, \quad n \geq 9
Specification 1
Strategy pack name: row_placements
Tree: http://permpal.com/tree/21259/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[14,x]*F[3,x]
F[3,x] = F[1,x]+F[38,x]+F[4,x]
F[4,x] = F[14,x]*F[5,x]
F[5,x] = F[1,x]+F[30,x]+F[32,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[14,x]*F[8,x]
F[8,x] = F[24,x]+F[9,x]
F[9,x] = F[10,x]+F[16,x]
F[10,x] = F[11,x]^2*F[15,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[14,x]
F[14,x] = x
F[15,x] = F[1,x]+F[14,x]
F[16,x] = F[17,x]
F[17,x] = F[14,x]*F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[14,x]*F[21,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[1,x]+F[19,x]
F[23,x] = F[11,x]*F[14,x]
F[24,x] = F[11,x]^2*F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[14,x]*F[27,x]
F[27,x] = F[22,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[11,x]*F[14,x]*F[27,x]
F[30,x] = F[14,x]*F[31,x]
F[31,x] = 2*F[30,x]+F[1,x]+F[6,x]
F[32,x] = F[14,x]*F[33,x]
F[33,x] = 2*F[30,x]+F[1,x]+F[32,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[14,x]*F[36,x]
F[36,x] = F[37,x]+F[8,x]
F[37,x] = F[11,x]^2*F[14,x]
F[38,x] = F[39,x]
F[39,x] = F[14,x]*F[40,x]
F[40,x] = F[3,x]+F[41,x]
F[41,x] = F[11,x]*F[14,x]*F[42,x]
F[42,x] = F[1,x]+F[43,x]+F[53,x]
F[43,x] = F[44,x]
F[44,x] = F[14,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]+F[52,x]
F[46,x] = F[1,x]+F[25,x]
F[47,x] = F[48,x]
F[48,x] = F[14,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[21,x]+F[23,x]
F[51,x] = F[11,x]*F[25,x]
F[52,x] = F[14,x]*F[45,x]
F[53,x] = F[54,x]
F[54,x] = F[11,x]*F[14,x]*F[46,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_{14}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{38}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{24}\! \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_{11} \left(x \right)^{2} F_{15}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{14}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{22}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{14}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = 2 F_{30}\! \left(x \right)+F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = 2 F_{30}\! \left(x \right)+F_{1}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{14}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{8}\! \left(x \right)
F_{37}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{14}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{14}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{3}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{1}\! \left(x \right)+F_{43}\! \left(x \right)+F_{53}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{14}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)+F_{52}\! \left(x \right)
F_{46}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{14}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{21}\! \left(x \right)+F_{23}\! \left(x \right)
F_{51}\! \left(x \right) = F_{11}\! \left(x \right) F_{25}\! \left(x \right)
F_{52}\! \left(x \right) = F_{14}\! \left(x \right) F_{45}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{46}\! \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_14(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_38(x) + F_4(x))
Eq(F_4(x), F_14(x)*F_5(x))
Eq(F_5(x), F_1(x) + F_30(x) + F_32(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_14(x)*F_8(x))
Eq(F_8(x), F_24(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_16(x))
Eq(F_10(x), F_11(x)**2*F_15(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_14(x))
Eq(F_14(x), x)
Eq(F_15(x), F_1(x) + F_14(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_14(x)*F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_14(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_1(x) + F_19(x))
Eq(F_23(x), F_11(x)*F_14(x))
Eq(F_24(x), F_11(x)**2*F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_14(x)*F_27(x))
Eq(F_27(x), F_22(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_11(x)*F_14(x)*F_27(x))
Eq(F_30(x), F_14(x)*F_31(x))
Eq(F_31(x), F_1(x) + 2*F_30(x) + F_6(x))
Eq(F_32(x), F_14(x)*F_33(x))
Eq(F_33(x), F_1(x) + 2*F_30(x) + F_32(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_14(x)*F_36(x))
Eq(F_36(x), F_37(x) + F_8(x))
Eq(F_37(x), F_11(x)**2*F_14(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_14(x)*F_40(x))
Eq(F_40(x), F_3(x) + F_41(x))
Eq(F_41(x), F_11(x)*F_14(x)*F_42(x))
Eq(F_42(x), F_1(x) + F_43(x) + F_53(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_14(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x) + F_52(x))
Eq(F_46(x), F_1(x) + F_25(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_14(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_21(x) + F_23(x))
Eq(F_51(x), F_11(x)*F_25(x))
Eq(F_52(x), F_14(x)*F_45(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_11(x)*F_14(x)*F_46(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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": "row_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, 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