0123_0132_0213_0231_1203_2301
Counting sequence:
1, 1, 2, 6, 18, 49, 121, 277, 601, 1258, 2574, 5196, 10412, 20791, 41463, 82679, 164931, 329192, 657396, 1313398, 2624894, 5247261, 10491237, 20978281, 41951293, 83896054, 167784106, 335558512, 671105376, 1342196883, 2684377379, 5368735531, 10737448647, 21474871316, 42949712688, 85899391034, 171798742866, 343597441177, 687194831921, 1374389606973, 2748779150049, 5497558228546, 10995116377222, 21990232665556, 43980465232468, 87960930355759, 175921860590991, 351843721049247, 703687441952651, 1407374883745408, 2814749767315884, 5629499534440766, 11258999068673382, 22517998137120341, 45035996273994813, 90071992547723089, 180143985095157701, 360287970190003662, 720575940379670946, 1441151880758979448, 2882303761517568904, 5764607523034718731, 11529215046068987707, 23058430092137493331, 46116860184274470543, 92233720368548389164, 184467440737096188776, 368934881474191748482, 737869762948382826426, 1475739525896764938833, 2951479051793529118089, 5902958103587057428901, 11805916207174114000617, 23611832414348227091866, 47223664828696453219838, 94447329657392905418844, 188894659314785809757436, 377789318629571618372647, 755578637259143235538471, 1511157274518286469802823, 3022314549036572938261459, 6044629098073145875105816, 12089258196146291748718692, 24178516392292583495865606, 48357032784585166990077518, 96714065569170333978416269, 193428131138340667955005461, 386856262276681335908092217, 773712524553362671814170701, 1547425049106725343626229158, 3094850098213450687250243994, 6189700196426901374498167936, 12379400392853802748993906352, 24758800785707605497985269891, 49517601571415210995967879763, 99035203142830421991932978299, 198070406285660843983863050071, 396140812571321687967723064132, 792281625142643375935442958496, 1584563250285286751870882609098, 3169126500570573503741761767714
Generating function in Maple syntax:
-(x^6+3*x^5-11*x^4+18*x^3-15*x^2+6*x-1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
-\frac{x^{6}+3 x^{5}-11 x^{4}+18 x^{3}-15 x^{2}+6 x -1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-x**6 - 3*x**5 + 11*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^5*F(x)+x^6+3*x^5-11*x^4+18*x^3-15*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+x^{6}+3 x^{5}-11 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-1+5*2^(-1+n)-37/12*n+11/24*n^2-5/12*n^3+1/24*n^4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -1+5 \,2^{-1+n}-\frac{37 n}{12}+\frac{11 n^{2}}{24}-\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) = 18
a(5) = 49
a(6) = 121
a(n+1) = 2*a(n)-1/24*(-1+n)*(n^3-13*n^2+22*n-48), 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) = 49
a \! \left(6\right) = 121
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(-1+n \right) \left(n^{3}-13 n^{2}+22 n -48\right)}{24}, \quad n \geq 7
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://www.permpal.com/tree/22052/
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[9,x]
F[3,x] = F[0,x]+F[4,x]
F[4,x] = F[5,x]
F[5,x] = F[6,x]*F[9,x]
F[6,x] = F[19,x]+F[29,x]+F[7,x]
F[7,x] = F[10,x]+F[8,x]
F[8,x] = F[1,x]+F[9,x]
F[9,x] = x
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[9,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[9,x]
F[17,x] = F[14,x]+F[9,x]
F[18,x] = F[11,x]*F[9,x]
F[19,x] = F[20,x]*F[9,x]
F[20,x] = F[21,x]+F[6,x]
F[21,x] = F[22,x]
F[22,x] = F[13,x]^2*F[23,x]*F[8,x]*F[9,x]
F[23,x] = F[13,x]+F[24,x]
F[24,x] = F[11,x]+F[25,x]
F[25,x] = F[15,x]+F[26,x]+F[28,x]
F[26,x] = F[27,x]*F[9,x]
F[27,x] = F[11,x]+F[25,x]
F[28,x] = F[24,x]*F[9,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[9,x]
F[31,x] = F[32,x]+F[39,x]
F[32,x] = F[13,x]*F[33,x]
F[33,x] = F[13,x]+F[34,x]
F[34,x] = F[11,x]+F[35,x]
F[35,x] = F[15,x]+F[36,x]+F[38,x]
F[36,x] = F[37,x]*F[9,x]
F[37,x] = F[11,x]+F[35,x]
F[38,x] = F[34,x]*F[9,x]
F[39,x] = F[13,x]*F[17,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_{9}\! \left(x \right)
F_{3}\! \left(x \right) = F_{0}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right) F_{9}\! \left(x \right)
F_{6}\! \left(x \right) = F_{19}\! \left(x \right)+F_{29}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{10}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{1}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = x
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \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) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{9}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right) F_{9}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{9}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{6}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13} \left(x \right)^{2} F_{23}\! \left(x \right) F_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{9}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{24}\! \left(x \right) F_{9}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{9}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{39}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{11}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{9}\! \left(x \right)
F_{37}\! \left(x \right) = F_{11}\! \left(x \right)+F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{34}\! \left(x \right) F_{9}\! \left(x \right)
F_{39}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \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)*F_9(x))
Eq(F_3(x), F_0(x) + F_4(x))
Eq(F_4(x), F_5(x))
Eq(F_5(x), F_6(x)*F_9(x))
Eq(F_6(x), F_19(x) + F_29(x) + F_7(x))
Eq(F_7(x), F_10(x) + F_8(x))
Eq(F_8(x), F_1(x) + F_9(x))
Eq(F_9(x), x)
Eq(F_10(x), F_11(x) + F_14(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_9(x))
Eq(F_13(x), F_1(x) + F_11(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_9(x))
Eq(F_17(x), F_14(x) + F_9(x))
Eq(F_18(x), F_11(x)*F_9(x))
Eq(F_19(x), F_20(x)*F_9(x))
Eq(F_20(x), F_21(x) + F_6(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_13(x)**2*F_23(x)*F_8(x)*F_9(x))
Eq(F_23(x), F_13(x) + F_24(x))
Eq(F_24(x), F_11(x) + F_25(x))
Eq(F_25(x), F_15(x) + F_26(x) + F_28(x))
Eq(F_26(x), F_27(x)*F_9(x))
Eq(F_27(x), F_11(x) + F_25(x))
Eq(F_28(x), F_24(x)*F_9(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_9(x))
Eq(F_31(x), F_32(x) + F_39(x))
Eq(F_32(x), F_13(x)*F_33(x))
Eq(F_33(x), F_13(x) + F_34(x))
Eq(F_34(x), F_11(x) + F_35(x))
Eq(F_35(x), F_15(x) + F_36(x) + F_38(x))
Eq(F_36(x), F_37(x)*F_9(x))
Eq(F_37(x), F_11(x) + F_35(x))
Eq(F_38(x), F_34(x)*F_9(x))
Eq(F_39(x), F_13(x)*F_17(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "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_and_col_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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [2, 3, 0, 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], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 2], [0, 2], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 2], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 0, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"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, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}, {"assumptions": [], "class_module": 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