0132_0213_1032_2301

Counting sequence:
1, 1, 2, 6, 20, 61, 168, 431, 1062, 2571, 6207, 15073, 36966, 91654, 229611, 580433, 1478327, 3788114, 9753653, 25208918, 65346421, 169778636, 441888312, 1151686916, 3004777505, 7845860611, 20499240448, 53584740496, 140120749742, 366509057211, 958869532102, 2509025775233, 6566060284376, 17184860081089, 44979929990421, 117737749982283, 308198960174032, 806790411013204, 2112034833855833, 5529039212584083, 14474533048011641, 37893460419743896, 99203649187876347, 259713089097276176, 679926822010821615, 1780049784749024586, 4660187347864031478, 12200441890098747286, 31940997584943696147, 83622269389755456601, 218925247634369062422, 573152347613444681018, 1500529543406151070136, 3928431779005380914249, 10284756786410736666356, 26925820565828319315683, 70492668882277202006362, 184552114023409248440519, 483163529072762467098363, 1264938184964502000754373, 3311650449359991231323786, 8670012010193967085921906, 22698383275378900812267607, 59425133204256716922978661, 155577007114019213101342475, 407305869691056848670909486, 1066340565065663185491655633, 2791715751718956412965182426, 7308806542517253463726766777, 19134703580684898798861534616, 50095303609241632574151380292, 131151206066448378206180447928, 343358312228920260609566449621, 898923725897945920752872725759, 2353412856020184535909760281916, 6161314823273141755497826187508, 16130531576020308867626555431442, 42230279829229921121467515483559, 110560307760553727044947342909618, 289450643150199805109717218227413, 757791621085582778476889723005356, 1983924218897622710706322774592541, 5193981033189433714412820249791217, 13598018875834975154073621274305871, 35600075584644085190891010173741460, 93202207858754467304765342449795936, 244006547952933690495736883583133413, 638817435922675351727109041116398791, 1672445759660349859774917705401563757, 4378519842748889217776299006361300756, 11463113767967347773911289176230460087, 30010821459915214064672188247428529588, 78569350609302414341534515016254341331, 205697230363040268802789835701735341494, 538522340469914871752551949889755912226, 1409869791026897305826299929569143505258, 3691087032571162964469215670020899596647, 9663391306607363425067082742900008168949, 25299086887092470985703503883492033706098, 66233869354353136881986371557201913702078, 173402521175333114360141496087365352204476

Generating function in Maple syntax:
-(6*x^7-30*x^6+62*x^5-82*x^4+69*x^3-34*x^2+9*x-1)/(2*x-1)/(x^2-3*x+1)/(-1+x)^5

Generating function in latex syntax:
-\frac{6 x^{7}-30 x^{6}+62 x^{5}-82 x^{4}+69 x^{3}-34 x^{2}+9 x -1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(-1+x \right)^{5}}

Generating function in sympy syntax:
(-6*x**7 + 30*x**6 - 62*x**5 + 82*x**4 - 69*x**3 + 34*x**2 - 9*x + 1)/((x - 1)**5*(2*x - 1)*(x**2 - 3*x + 1))

Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(-1+x)^5*F(x)+6*x^7-30*x^6+62*x^5-82*x^4+69*x^3-34*x^2+9*x-1 = 0

Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(-1+x \right)^{5} F \! \left(x \right)+6 x^{7}-30 x^{6}+62 x^{5}-82 x^{4}+69 x^{3}-34 x^{2}+9 x -1 = 0

Explicit closed form in Maple syntax:
1/120*(-12*5^(1/2)+60)*(3/2-1/2*5^(1/2))^(-n)+1/120*(12*5^(1/2)+60)*(3/2+1/2*5^(1/2))^(-n)+1/24*n^4-5/12*n^3-1/24*n^2-19/12*n+2^(n+1)-2

Explicit closed form in latex syntax:
\frac{\left(-12 \sqrt{5}+60\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{\left(12 \sqrt{5}+60\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{n^{4}}{24}-\frac{5 n^{3}}{12}-\frac{n^{2}}{24}-\frac{19 n}{12}+2^{n +1}-2

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 61
a(6) = 168
a(7) = 431
a(n+3) = 1/24*n^4-5/12*n^3-25/24*n^2+2*a(n)-7*a(n+1)+5*a(n+2)+29/12*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) = 61
a \! \left(6\right) = 168
a \! \left(7\right) = 431
a \! \left(n +3\right) = \frac{n^{4}}{24}-\frac{5 n^{3}}{12}-\frac{25 n^{2}}{24}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+\frac{29 n}{12}+1, \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21184/
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[14,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[15,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[14,x]*F[8,x]
F[8,x] = F[11,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[6,x]
F[11,x] = F[12,x]*F[6,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[12,x]*F[14,x]
F[14,x] = x
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[17,x]
F[17,x] = F[18,x]+F[5,x]
F[18,x] = F[19,x]
F[19,x] = F[14,x]*F[20,x]
F[20,x] = F[21,x]+F[38,x]
F[21,x] = F[22,x]+F[37,x]
F[22,x] = F[23,x]+F[35,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[1,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[14,x]*F[25,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[13,x]
F[30,x] = F[31,x]+F[32,x]+F[34,x]
F[31,x] = 0
F[32,x] = F[14,x]*F[33,x]
F[33,x] = F[26,x]+F[30,x]
F[34,x] = F[14,x]*F[28,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]^2*F[14,x]*F[26,x]
F[37,x] = F[12,x]*F[29,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[26,x]+F[40,x]
F[40,x] = F[31,x]+F[41,x]+F[42,x]
F[41,x] = F[14,x]*F[39,x]
F[42,x] = F[14,x]*F[43,x]
F[43,x] = F[29,x]+F[40,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_{14}\! \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_{15}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{11}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{6}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{6}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{5}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{38}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{37}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{35}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right) F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{14}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{26}\! \left(x \right)+F_{30}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right) F_{28}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12} \left(x \right)^{2} F_{14}\! \left(x \right) F_{26}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{26}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{31}\! \left(x \right)+F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{14}\! \left(x \right) F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{14}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{29}\! \left(x \right)+F_{40}\! \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_14(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_15(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_14(x)*F_8(x))
Eq(F_8(x), F_11(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_6(x))
Eq(F_11(x), F_12(x)*F_6(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_12(x)*F_14(x))
Eq(F_14(x), x)
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_5(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_14(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_38(x))
Eq(F_21(x), F_22(x) + F_37(x))
Eq(F_22(x), F_23(x) + F_35(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_1(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_14(x)*F_25(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_13(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_34(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_14(x)*F_33(x))
Eq(F_33(x), F_26(x) + F_30(x))
Eq(F_34(x), F_14(x)*F_28(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)**2*F_14(x)*F_26(x))
Eq(F_37(x), F_12(x)*F_29(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_26(x) + F_40(x))
Eq(F_40(x), F_31(x) + F_41(x) + F_42(x))
Eq(F_41(x), F_14(x)*F_39(x))
Eq(F_42(x), F_14(x)*F_43(x))
Eq(F_43(x), F_29(x) + F_40(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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 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