0132_1320_2103_2130
Counting sequence:
1, 1, 2, 6, 20, 63, 181, 481, 1206, 2899, 6759, 15409, 34544, 76451, 167497, 364001, 785770, 1686755, 3603531, 7666593, 16251620, 34339347, 72349997, 152041521, 318764830, 666891763, 1392506031, 2902455121, 6039794136, 12549353539, 26038234769, 53955521729, 111669144274, 230854486211, 476741363347, 983547503681, 2027224555980, 4174708203315, 8589934582901, 17660905511313, 36283883705990, 74491912770387, 152832116248567, 313360813902961, 642114790607040, 1315015906804963, 2691604464779481, 5506354231884961, 11258999068407866, 23010579346076579, 47006321110658779, 95982967058311649, 195906583790593204, 399694466929106771, 815151532554033597, 1661828262499685361, 3386706919782583790, 6899514629131569075, 14051230837395915071, 28606864833057356433, 58222535982645736360, 118462684598353489027, 240960594462830978337, 489991639457909923201, 996124179980315743650, 2024530162089623244163, 4113623928437229962531, 8356375065390426832001, 16971004547812787434396, 34458517929689442364019, 69950053527506619670789, 141966142391268709177169, 288064355455048357973334, 584392852255118595130131, 1185313987200280948570247, 2403684539780649413701041, 4873482210321473860461200, 9879190682163297786976035, 20022833887367295705992041, 40574572820815991675993953, 82206955733794783879934730, 166529531651915168815687267, 337290303672481539742931307, 683043088082265483708894241, 1383011137639135775863766660, 2799872198227481168619401363, 5667444242353381571022447181, 11470288176503601609612088241, 23211375736600880154358465726, 46964350240389114178985407859, 95011898015152936098507662799, 192190191099055287678088910289, 388713172335609406318324876664, 786091924946216474560943748291, 1589515010442428272970475365297, 3213692341984847193638126342721, 6496709326169675682670603780210, 13132067936739313956129909616195, 26541434442278553093837223205811, 53637466022156956550829254215873, 108384126319513613827968123893100
Generating function in Maple syntax:
(x^7-x^6-8*x^5+21*x^4-28*x^3+20*x^2-7*x+1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
\frac{x^{7}-x^{6}-8 x^{5}+21 x^{4}-28 x^{3}+20 x^{2}-7 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**7 - x**6 - 8*x**5 + 21*x**4 - 28*x**3 + 20*x**2 - 7*x + 1)/((x - 1)**4*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^2*(x-1)^4*F(x)+x^7-x^6-8*x^5+21*x^4-28*x^3+20*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)^{4} F \! \left(x \right)+x^{7}-x^{6}-8 x^{5}+21 x^{4}-28 x^{3}+20 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,2+7/8*2^n*n-2^(n+1)+7/6*n-1/6*n^3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2+\frac{7 \,2^{n} n}{8}-2^{n +1}+\frac{7 n}{6}-\frac{n^{3}}{6} & \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) = 181
a(7) = 481
a(n+2) = -1/6*n^3+n^2-4*a(n)+4*a(n+1)+7/6*n-1, n >= 8
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) = 181
a \! \left(7\right) = 481
a \! \left(n +2\right) = -\frac{n^{3}}{6}+n^{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{7 n}{6}-1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15669/
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[22,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]+F[16,x]+F[21,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[4,x]
F[21,x] = F[13,x]*F[4,x]
F[22,x] = F[2,x]+F[23,x]
F[23,x] = F[15,x]+F[24,x]+F[39,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]+F[7,x]
F[27,x] = F[15,x]+F[16,x]+F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[13,x]+F[30,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[16,x]
F[32,x] = 2*F[15,x]+F[33,x]+F[38,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]
F[35,x] = F[19,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[35,x]*F[4,x]
F[38,x] = F[30,x]*F[4,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[47,x]
F[41,x] = F[2,x]+F[42,x]
F[42,x] = F[15,x]+F[43,x]+F[46,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[11,x]+F[14,x]
F[46,x] = F[4,x]*F[41,x]
F[47,x] = F[48,x]+F[52,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[31,x]+F[7,x]
F[52,x] = 2*F[15,x]+F[53,x]+F[56,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[14,x]+F[32,x]
F[56,x] = F[4,x]*F[47,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_{22}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{21}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{39}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{7}\! \left(x \right)
F_{27}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{16}\! \left(x \right)
F_{32}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{38}\! \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_{19}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{47}\! \left(x \right)
F_{41}\! \left(x \right) = F_{2}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{46}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{52}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{31}\! \left(x \right)+F_{7}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{53}\! \left(x \right)+F_{56}\! \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_{14}\! \left(x \right)+F_{32}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \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_22(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_21(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_4(x))
Eq(F_21(x), F_13(x)*F_4(x))
Eq(F_22(x), F_2(x) + F_23(x))
Eq(F_23(x), F_15(x) + F_24(x) + F_39(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x) + F_7(x))
Eq(F_27(x), F_15(x) + F_16(x) + F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_13(x) + F_30(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_16(x))
Eq(F_32(x), 2*F_15(x) + F_33(x) + F_38(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_19(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_35(x)*F_4(x))
Eq(F_38(x), F_30(x)*F_4(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_47(x))
Eq(F_41(x), F_2(x) + F_42(x))
Eq(F_42(x), F_15(x) + F_43(x) + F_46(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_11(x) + F_14(x))
Eq(F_46(x), F_4(x)*F_41(x))
Eq(F_47(x), F_48(x) + F_52(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_31(x) + F_7(x))
Eq(F_52(x), 2*F_15(x) + F_53(x) + F_56(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_14(x) + F_32(x))
Eq(F_56(x), F_4(x)*F_47(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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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