0213_0231_0321_2301_3120

Counting sequence:
1, 1, 2, 6, 19, 51, 122, 272, 581, 1209, 2476, 5022, 10127, 20351, 40814, 81756, 163657, 327477, 655136, 1310474, 2621171, 5242587, 10485442, 20971176, 41942669, 83885681, 167771732, 335543862, 671088151, 1342176759, 2684354006, 5368708532, 10737417617, 21474835821, 42949672264, 85899345186, 171798691067, 343597382867, 687194766506, 1374389533824, 2748779068501, 5497558137897, 10995116276732, 21990232554446, 43980465109919, 87960930220911, 175921860442942, 351843720887052, 703687441775321, 1407374883551909, 2814749767105136, 5629499534211642, 11258999068424707, 22517998136850891, 45035996273703314, 90071992547408216, 180143985094818077, 360287970189637857, 720575940379277476, 1441151880758556774, 2882303761517115431, 5764607523034232807, 11529215046068467622, 23058430092136937316, 46116860184273876769, 92233720368547755741, 184467440737095513752, 368934881474191029842, 737869762948382062091, 1475739525896764126659, 2951479051793528255866, 5902958103587056514352, 11805916207174113031397, 23611832414348226065561, 47223664828696452133964, 94447329657392904270846, 188894659314785808544687, 377789318629571617092447, 755578637259143234188046, 1511157274518286468379324, 3022314549036572936761961, 6044629098073145873527317, 12089258196146291747058112, 24178516392292583494119786, 48357032784585166988243219, 96714065569170333976490171, 193428131138340667952984162, 386856262276681335905972232, 773712524553362671811948461, 1547425049106725343623901009, 3094850098213450687247806196, 6189700196426901374495616662, 12379400392853802748991237687, 24758800785707605497982479831, 49517601571415210995964964214, 99035203142830421991929933076, 198070406285660843983859870897, 396140812571321687967719746637, 792281625142643375935439498216, 1584563250285286751870879001474, 3169126500570573503741758008091

Generating function in Maple syntax:
-(2*x^5-2*x^4+2*x^3-6*x^2+4*x-1)/(2*x-1)/(x-1)^3

Generating function in latex syntax:
-\frac{2 x^{5}-2 x^{4}+2 x^{3}-6 x^{2}+4 x -1}{\left(2 x -1\right) \left(x -1\right)^{3}}

Generating function in sympy syntax:
(-2*x**5 + 2*x**4 - 2*x**3 + 6*x**2 - 4*x + 1)/((x - 1)**3*(2*x - 1))

Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^3*F(x)+2*x^5-2*x^4+2*x^3-6*x^2+4*x-1 = 0

Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+2 x^{5}-2 x^{4}+2 x^{3}-6 x^{2}+4 x -1 = 0

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1+5*2^(-1+n)-7/2*n-1/2*n^2)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 1+5 \,2^{-1+n}-\frac{7 n}{2}-\frac{n^{2}}{2} & \mathit{\text{otherwise}} \end{array}\right.

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(n+1) = 1/2*n^2+2*a(n)+5/2*n-5, n >= 6

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) = 19
a \! \left(5\right) = 51
a \! \left(n +1\right) = \frac{n^{2}}{2}+2 a \! \left(n \right)+\frac{5 n}{2}-5, \quad n \geq 6

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21834/
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[18,x]*F[4,x]
F[4,x] = F[21,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[18,x]*F[9,x]
F[9,x] = F[10,x]+F[5,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[16,x]*F[18,x]
F[12,x] = F[13,x]+F[19,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]*F[18,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[16,x]*F[18,x]
F[18,x] = x
F[19,x] = F[18,x]*F[20,x]
F[20,x] = 2*F[19,x]+F[1,x]
F[21,x] = F[22,x]
F[22,x] = F[18,x]*F[23,x]
F[23,x] = F[24,x]+F[41,x]
F[24,x] = F[25,x]+F[34,x]
F[25,x] = F[26,x]+F[32,x]
F[26,x] = F[13,x]*F[27,x]
F[27,x] = F[1,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]^2*F[18,x]
F[30,x] = F[1,x]+F[31,x]
F[31,x] = F[18,x]*F[30,x]
F[32,x] = F[16,x]*F[30,x]*F[33,x]
F[33,x] = F[19,x]
F[34,x] = F[30,x]*F[35,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[37,x]+F[7,x]
F[37,x] = F[33,x]*F[38,x]
F[38,x] = F[17,x]
F[39,x] = F[38,x]*F[40,x]
F[40,x] = F[14,x]
F[41,x] = F[42,x]
F[42,x] = F[16,x]*F[18,x]*F[40,x]*F[43,x]
F[43,x] = F[31,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_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{21}\! \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_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{18}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{5}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right) F_{18}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{19}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{18}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{16}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = x
F_{19}\! \left(x \right) = F_{18}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{1}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{41}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{34}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{32}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30} \left(x \right)^{2} F_{18}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{18}\! \left(x \right) F_{30}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right) F_{30}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{19}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{7}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{17}\! \left(x \right)
F_{39}\! \left(x \right) = F_{38}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{14}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{16}\! \left(x \right) F_{18}\! \left(x \right) F_{40}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{31}\! \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_18(x)*F_4(x))
Eq(F_4(x), F_21(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_8(x))
Eq(F_8(x), F_18(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_5(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_16(x)*F_18(x))
Eq(F_12(x), F_13(x) + F_19(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x)*F_18(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_16(x)*F_18(x))
Eq(F_18(x), x)
Eq(F_19(x), F_18(x)*F_20(x))
Eq(F_20(x), F_1(x) + 2*F_19(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_18(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_41(x))
Eq(F_24(x), F_25(x) + F_34(x))
Eq(F_25(x), F_26(x) + F_32(x))
Eq(F_26(x), F_13(x)*F_27(x))
Eq(F_27(x), F_1(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_18(x)*F_30(x)**2)
Eq(F_30(x), F_1(x) + F_31(x))
Eq(F_31(x), F_18(x)*F_30(x))
Eq(F_32(x), F_16(x)*F_30(x)*F_33(x))
Eq(F_33(x), F_19(x))
Eq(F_34(x), F_30(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_37(x) + F_7(x))
Eq(F_37(x), F_33(x)*F_38(x))
Eq(F_38(x), F_17(x))
Eq(F_39(x), F_38(x)*F_40(x))
Eq(F_40(x), F_14(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_16(x)*F_18(x)*F_40(x)*F_43(x))
Eq(F_43(x), F_31(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, 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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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, 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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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, 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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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": "RequirementInsertionStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[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], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[2, 0], 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