0132_0231_0321_2031_3021_3120
Counting sequence:
1, 1, 2, 6, 18, 50, 129, 315, 740, 1692, 3795, 8393, 18366, 39858, 85925, 184215, 393096, 835448, 1769319, 3735381, 7864130, 16514862, 34602777, 72351491, 150994668, 314572500, 654311099, 1358954145, 2818571910, 5838470762, 12079595085, 24964496943, 51539607056, 106300440048, 219043331535, 450971565485, 927712935306, 1906965478758, 3917010173249, 8040178777371, 16492674415860, 33809982553292, 69269232549027, 141836999982201, 290271069731918, 593736278998050, 1213860837063669, 2480498232261575, 5066549580790680, 10344205394115432, 21110623253297975, 43065671436729093, 87820192733723346, 179018085187975838, 364791569817008745, 743093938516130355, 1513209474796485116, 3080462145121417668, 6269010681299728779, 12754194144713242961, 25940733853654055190, 52746158835763247322, 107221699928436766877, 217902164370694076511, 442721857769029236768, 899278773593340639200, 1826227663297245607839, 3707795558815619872605, 7526271582073497057050, 15273904093031508735702, 30990530043832046712465, 62866503803202151904843, 127503895037480420767236, 258549564937113075447228, 524182679598530618717555, 1062532458645670173078825, 2153399116188558217442526, 4363466630171552177452178, 8840270055931975840035909, 17907213703041694650332151, 36267774588438875241182120, 73442243541588722363396952, 148697875812599388488856327, 301022529084042664501834421, 609298613085773104051909218, 1233104336006921758200295950, 2495222891684594616593543609, 5048474222710691433572987235, 10213005324104387267917771020, 20658124405574783337379131572, 41780476325881584277845438555, 84489407681227203761865224193, 170835725421382477936079138726, 345385270960621096696855654218, 698198182156954475043106057965, 1411251644785333513385001610895, 2852213850513516153367582207536, 5763848822912730559930322382288, 11646539889596857626250960694639, 23530764266736508265282553244941, 47536897508558602556126370196650
Generating function in Maple syntax:
(x^6-5*x^4+14*x^3-14*x^2+6*x-1)/(2*x-1)^2/(-1+x)^3
Generating function in latex syntax:
\frac{x^{6}-5 x^{4}+14 x^{3}-14 x^{2}+6 x -1}{\left(2 x -1\right)^{2} \left(-1+x \right)^{3}}
Generating function in sympy syntax:
(x**6 - 5*x**4 + 14*x**3 - 14*x**2 + 6*x - 1)/((x - 1)**3*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^2*(-1+x)^3*F(x)+x^6-5*x^4+14*x^3-14*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{2} \left(-1+x \right)^{3} F \! \left(x \right)+x^{6}-5 x^{4}+14 x^{3}-14 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/8*n*(3*2^n-4*n+4))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{n \left(3 \,2^{n}-4 n +4\right)}{8} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 50
a(6) = 129
a(n+2) = -1/2*n^2-4*a(n)+4*a(n+1)+5/2*n-1, 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) = 18
a \! \left(5\right) = 50
a \! \left(6\right) = 129
a \! \left(n +2\right) = -\frac{n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{5 n}{2}-1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22352/
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[13,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[25,x]+F[8,x]
F[8,x] = F[19,x]*F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[9,x]
F[13,x] = x
F[14,x] = F[11,x]+F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[13,x]*F[18,x]
F[18,x] = F[1,x]+F[17,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[13,x]*F[19,x]*F[22,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[13,x]*F[22,x]
F[25,x] = F[16,x]+F[26,x]
F[26,x] = F[27,x]+F[28,x]+F[31,x]
F[27,x] = 0
F[28,x] = F[13,x]*F[29,x]
F[29,x] = F[23,x]+F[30,x]
F[30,x] = F[28,x]
F[31,x] = F[13,x]*F[32,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[16,x]+F[34,x]
F[34,x] = F[27,x]+F[28,x]+F[35,x]
F[35,x] = F[13,x]*F[33,x]
F[36,x] = F[37,x]+F[43,x]
F[37,x] = F[27,x]+F[38,x]+F[41,x]
F[38,x] = F[13,x]*F[39,x]
F[39,x] = F[13,x]+F[40,x]
F[40,x] = F[38,x]
F[41,x] = F[13,x]*F[42,x]
F[42,x] = F[16,x]+F[37,x]
F[43,x] = F[27,x]+F[44,x]+F[49,x]+F[51,x]
F[44,x] = F[13,x]*F[45,x]
F[45,x] = F[46,x]+F[48,x]
F[46,x] = F[47,x]
F[47,x] = F[13,x]*F[23,x]
F[48,x] = F[44,x]
F[49,x] = F[13,x]*F[50,x]
F[50,x] = F[34,x]+F[43,x]
F[51,x] = F[13,x]*F[36,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{25}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{19}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \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_{13}\! \left(x \right) F_{9}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{11}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right) F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{13}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{23}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{16}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{13}\! \left(x \right) F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{43}\! \left(x \right)
F_{37}\! \left(x \right) = F_{27}\! \left(x \right)+F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{13}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{13}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{16}\! \left(x \right)+F_{37}\! \left(x \right)
F_{43}\! \left(x \right) = F_{27}\! \left(x \right)+F_{44}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)
F_{44}\! \left(x \right) = F_{13}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{13}\! \left(x \right) F_{23}\! \left(x \right)
F_{48}\! \left(x \right) = F_{44}\! \left(x \right)
F_{49}\! \left(x \right) = F_{13}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{34}\! \left(x \right)+F_{43}\! \left(x \right)
F_{51}\! \left(x \right) = F_{13}\! \left(x \right) F_{36}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_13(x)*F_7(x))
Eq(F_7(x), F_25(x) + F_8(x))
Eq(F_8(x), F_19(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_9(x))
Eq(F_13(x), x)
Eq(F_14(x), F_11(x) + F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_13(x)*F_18(x))
Eq(F_18(x), F_1(x) + F_17(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_13(x)*F_19(x)*F_22(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_13(x)*F_22(x))
Eq(F_25(x), F_16(x) + F_26(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_31(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_13(x)*F_29(x))
Eq(F_29(x), F_23(x) + F_30(x))
Eq(F_30(x), F_28(x))
Eq(F_31(x), F_13(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_16(x) + F_34(x))
Eq(F_34(x), F_27(x) + F_28(x) + F_35(x))
Eq(F_35(x), F_13(x)*F_33(x))
Eq(F_36(x), F_37(x) + F_43(x))
Eq(F_37(x), F_27(x) + F_38(x) + F_41(x))
Eq(F_38(x), F_13(x)*F_39(x))
Eq(F_39(x), F_13(x) + F_40(x))
Eq(F_40(x), F_38(x))
Eq(F_41(x), F_13(x)*F_42(x))
Eq(F_42(x), F_16(x) + F_37(x))
Eq(F_43(x), F_27(x) + F_44(x) + F_49(x) + F_51(x))
Eq(F_44(x), F_13(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_48(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_13(x)*F_23(x))
Eq(F_48(x), F_44(x))
Eq(F_49(x), F_13(x)*F_50(x))
Eq(F_50(x), F_34(x) + F_43(x))
Eq(F_51(x), F_13(x)*F_36(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": 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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": 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