0132_0231_2301_3012
Counting sequence:
1, 1, 2, 6, 20, 64, 190, 526, 1376, 3444, 8330, 19618, 45244, 102616, 229622, 508182, 1114424, 2425180, 5243266, 11272618, 24117716, 51380736, 109052462, 230687326, 486539920, 1023410884, 2147484410, 4496294706, 9395241836, 19595789224, 40802190310, 84825605158, 176093660264, 365072221356, 755914245362, 1563368097082, 3229815408004, 6665789244880, 13743895348766, 28312424416878, 58274116273856, 119846767429396, 246290604623722, 505775348778946, 1037938976622620, 2128654511376504, 4362862139017430, 8936830510565686, 18295873486195096, 37436171902519804, 76561193665301090, 156500087051127498, 319755573543308084, 653021945968724896, 1333065489701669902, 2720174174931782782, 5548434740920454384, 11313042263954689380, 23058430092136943066, 46981551312729017938, 95692484882368302796, 194843734278557142856, 396604997584755363782, 807045053224792887366, 1641760222560150098120, 3338860677341428846924, 6788401819125114999250, 13798164567134744613466, 28039050992038518461156, 56963545699615095395184, 115697978830306307740670, 234937732522764849386638, 476959014769834166588704, 968085128988277268813236, 1964504456873772408903242, 3985677311541980560365282, 8084691418672832605853564, 16396056428523408181958680, 33245460039402302304426166, 67397614443515576489875798, 136608617616453096741804536, 276844012691750081007721116, 560941580301187937063672642, 1136390270437751424223812586, 2301794760546253948640566420, 4661617960434010097667022144, 9439292799551024596105829870, 19110699356468057993755238046, 38685626227668133590597640016, 78299707484800302387369615364, 158456325028528675187087909050, 320626470174913491198873182578, 648680580585539264047141102124, 1312216441642503091393071686376, 2654143444227855309383722345382, 5367708010341408871962602644582, 10854258264454214250315521205544, 21946201016451221513411674252780, 44367771007988029052384612198066, 89686279966147230155891751790458, 181274035832636804414028558379076
Generating function in Maple syntax:
-(7*x^4-14*x^3+14*x^2-6*x+1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
-\frac{7 x^{4}-14 x^{3}+14 x^{2}-6 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-7*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*(x-1)^3*F(x)+7*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(x -1\right)^{3} F \! \left(x \right)+7 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
8+n^2+3*n+3*2^(-1+n)*n-7*2^n
Explicit closed form in latex syntax:
8+n^{2}+3 n +3 \,2^{-1+n} n -7 \,2^{n}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+2) = n^2-4*a(n)+4*a(n+1)-n+2, n >= 5
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(n +2\right) = n^{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-n +2, \quad n \geq 5
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/21252/
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[4,x]
F[3,x] = x
F[4,x] = F[28,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[3,x]*F[7,x]
F[7,x] = F[5,x]+F[8,x]
F[8,x] = F[3,x]*F[9,x]
F[9,x] = F[10,x]+F[12,x]+F[8,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[10,x]*F[3,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[14,x]*F[3,x]
F[14,x] = F[15,x]+F[23,x]
F[15,x] = F[10,x]+F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[11,x]
F[18,x] = F[19,x]+F[20,x]+F[22,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[3,x]
F[21,x] = F[17,x]+F[18,x]
F[22,x] = F[16,x]*F[3,x]
F[23,x] = F[17,x]+F[24,x]
F[24,x] = F[19,x]+F[25,x]+F[26,x]
F[25,x] = F[23,x]*F[3,x]
F[26,x] = F[27,x]*F[3,x]
F[27,x] = F[17,x]+F[24,x]
F[28,x] = F[29,x]
F[29,x] = F[3,x]*F[30,x]
F[30,x] = F[31,x]+F[37,x]+F[4,x]
F[31,x] = F[32,x]
F[32,x] = F[3,x]*F[33,x]
F[33,x] = F[30,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[10,x]^2*F[3,x]*F[36,x]
F[36,x] = F[10,x]+F[27,x]
F[37,x] = F[3,x]*F[38,x]
F[38,x] = F[39,x]+F[43,x]+F[44,x]
F[39,x] = F[10,x]+F[40,x]
F[40,x] = F[17,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[3,x]*F[40,x]
F[43,x] = F[3,x]*F[38,x]
F[44,x] = F[45,x]
F[45,x] = F[10,x]*F[3,x]*F[46,x]
F[46,x] = F[21,x]+F[47,x]
F[47,x] = F[10,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_{4}\! \left(x \right)
F_{3}\! \left(x \right) = x
F_{4}\! \left(x \right) = F_{28}\! \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_{3}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{5}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{3}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{12}\! \left(x \right)+F_{8}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{10}\! \left(x \right) F_{3}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right) F_{3}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)
F_{15}\! \left(x \right) = F_{10}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{22}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{3}\! \left(x \right)
F_{21}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right) F_{3}\! \left(x \right)
F_{23}\! \left(x \right) = F_{17}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{19}\! \left(x \right)+F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{23}\! \left(x \right) F_{3}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{3}\! \left(x \right)
F_{27}\! \left(x \right) = F_{17}\! \left(x \right)+F_{24}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{3}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{37}\! \left(x \right)+F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{3}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{10} \left(x \right)^{2} F_{3}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right)+F_{27}\! \left(x \right)
F_{37}\! \left(x \right) = F_{3}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{17}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{3}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{3}\! \left(x \right) F_{38}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{10}\! \left(x \right) F_{3}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{21}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{10}\! \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)*F_4(x))
Eq(F_3(x), x)
Eq(F_4(x), F_28(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_3(x)*F_7(x))
Eq(F_7(x), F_5(x) + F_8(x))
Eq(F_8(x), F_3(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_12(x) + F_8(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_10(x)*F_3(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_14(x)*F_3(x))
Eq(F_14(x), F_15(x) + F_23(x))
Eq(F_15(x), F_10(x) + F_16(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_11(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_22(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_3(x))
Eq(F_21(x), F_17(x) + F_18(x))
Eq(F_22(x), F_16(x)*F_3(x))
Eq(F_23(x), F_17(x) + F_24(x))
Eq(F_24(x), F_19(x) + F_25(x) + F_26(x))
Eq(F_25(x), F_23(x)*F_3(x))
Eq(F_26(x), F_27(x)*F_3(x))
Eq(F_27(x), F_17(x) + F_24(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_3(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_37(x) + F_4(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_3(x)*F_33(x))
Eq(F_33(x), F_30(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_10(x)**2*F_3(x)*F_36(x))
Eq(F_36(x), F_10(x) + F_27(x))
Eq(F_37(x), F_3(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_43(x) + F_44(x))
Eq(F_39(x), F_10(x) + F_40(x))
Eq(F_40(x), F_17(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_3(x)*F_40(x))
Eq(F_43(x), F_3(x)*F_38(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_10(x)*F_3(x)*F_46(x))
Eq(F_46(x), F_21(x) + F_47(x))
Eq(F_47(x), F_10(x)*F_36(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, 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": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[1, 2], [1, 0], [1, 2], [1, 2]]}, {"patt": [3, 0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}], "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": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 2, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"children": [{"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]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 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