0132_0231_1032_2301

Counting sequence:
1, 1, 2, 6, 20, 64, 196, 583, 1702, 4906, 14010, 39716, 111912, 313741, 875690, 2434734, 6746336, 18636328, 51340972, 141089491, 386857870, 1058572306, 2891193222, 7882921676, 21458980560, 58330331929, 158339542226, 429274563798, 1162435429292, 3144299295376, 8496313890580, 22935929106271, 61859482184182, 166695256984186, 448836498626130, 1207592128929524, 3246663348410616, 8722790407011493, 23420161884503162, 62842664791426686, 168524287112683640, 451674590870415736, 1209916213847031292, 3239379841392289771, 8668723954146205150, 23186988161773815394, 61992328309541343006, 165670063961221029788, 442557977378868092832, 1181744142395251471921, 3154355158661744646050, 8416623185934689212326, 22449739247321584906052, 59859967248223144505056, 159558055725747823691620, 425170506922027172818039, 1132594492790954258424646, 3016179747709690447194058, 8030004051364061053709802, 21372443533201469780853956, 56869100627671335519055560, 151281569461286521400004541, 402333967011179195321829962, 1069748698225749979864409166, 2843638868371576523527297040, 7557319762351998013186256584, 20080049244367967004181897036, 53341862592339533041341932803, 141670913571729972757640971822, 376187968618500777102989695666, 998718888731644193527757183670, 2650929296424396916538160347756, 7035124900637570060283930268272, 18666652506928418663847371190601, 49520398024371978625662199095410, 131349030677420289896817532737654, 348334595937362916421421303251676, 923623973811857762753660782777648, 2448637073600304256641573358227892, 6490617274618147358190881825585551, 17202105085039320986188527406420630, 45584038957226272754127243723672346, 120776144382033685569193378466692482, 319954450998330891678700872452697236, 847491246445933124349653006518179480, 2244521345028934778293241469205889749, 5943674920876295966416277599784855066, 15737307757616761091691226604100722718, 41663059239788986464978101835688089512, 110285498285178387801469542498523827736, 291899509698215746277789216823393462940, 772497624733449369548747717866626486811, 2044141072191604348580644075297885705342, 5408484120985799116312905313696259826946, 14308439170509627334505044144277181660750, 37849658500630150449523423148924919613244, 100111883781898192366881841113380192667456, 264767210086529464147250751594073870396513, 700162050751567944210439752066533439055682, 1851354637748342563394647868395744296363846, 4894836644960086586569672606093660978282100

Generating function in Maple syntax:
(x^6-4*x^5+10*x^4-20*x^3+18*x^2-7*x+1)/(x-1)^2/(x^2-3*x+1)^2

Generating function in latex syntax:
\frac{x^{6}-4 x^{5}+10 x^{4}-20 x^{3}+18 x^{2}-7 x +1}{\left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2}}

Generating function in sympy syntax:
(x**6 - 4*x**5 + 10*x**4 - 20*x**3 + 18*x**2 - 7*x + 1)/((x - 1)**2*(x**2 - 3*x + 1)**2)

Implicit equation for the generating function in Maple syntax:
(x-1)^2*(x^2-3*x+1)^2*F(x)-x^6+4*x^5-10*x^4+20*x^3-18*x^2+7*x-1 = 0

Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)-x^{6}+4 x^{5}-10 x^{4}+20 x^{3}-18 x^{2}+7 x -1 = 0

Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/50*((-5*n+26)*5^(1/2)+15*n-50)*(3/2-1/2*5^(1/2))^(-n)+1/50*((5*n-26)*5^(1/2)+15*n-50)*(3/2+1/2*5^(1/2))^(-n)-n+2)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-5 n +26\right) \sqrt{5}+15 n -50\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{50}+\\\frac{\left(\left(5 n -26\right) \sqrt{5}+15 n -50\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{50}-n +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) = 20
a(5) = 64
a(6) = 196
a(n+4) = -a(n)+6*a(n+1)-11*a(n+2)+6*a(n+3)-n, 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) = 20
a \! \left(5\right) = 64
a \! \left(6\right) = 196
a \! \left(n +4\right) = -a \! \left(n \right)+6 a \! \left(n +1\right)-11 a \! \left(n +2\right)+6 a \! \left(n +3\right)-n, \quad n \geq 7

Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/21226/
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[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[3,x]*F[7,x]
F[7,x] = F[22,x]+F[28,x]+F[8,x]
F[8,x] = F[0,x]+F[9,x]
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[3,x]
F[11,x] = F[12,x]+F[16,x]
F[12,x] = F[0,x]*F[13,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[3,x]
F[16,x] = F[13,x]*F[17,x]
F[17,x] = F[18,x]+F[3,x]
F[18,x] = F[19,x]+F[20,x]+F[21,x]
F[19,x] = 0
F[20,x] = F[14,x]*F[3,x]
F[21,x] = F[17,x]*F[3,x]
F[22,x] = F[23,x]
F[23,x] = F[13,x]*F[24,x]*F[3,x]
F[24,x] = F[13,x]+F[25,x]+F[26,x]
F[25,x] = F[24,x]*F[3,x]
F[26,x] = F[27,x]
F[27,x] = F[13,x]*F[24,x]*F[3,x]
F[28,x] = F[29,x]
F[29,x] = F[13,x]*F[3,x]*F[7,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_{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_{3}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{22}\! \left(x \right)+F_{28}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{0}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{3}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{0}\! \left(x \right) F_{13}\! \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_{3}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{3}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{21}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{3}\! \left(x \right)
F_{21}\! \left(x \right) = F_{17}\! \left(x \right) F_{3}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right) F_{24}\! \left(x \right) F_{3}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right)+F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{24}\! \left(x \right) F_{3}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right) F_{24}\! \left(x \right) F_{3}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right) F_{3}\! \left(x \right) F_{7}\! \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_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_3(x)*F_7(x))
Eq(F_7(x), F_22(x) + F_28(x) + F_8(x))
Eq(F_8(x), F_0(x) + F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_3(x))
Eq(F_11(x), F_12(x) + F_16(x))
Eq(F_12(x), F_0(x)*F_13(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_3(x))
Eq(F_16(x), F_13(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_3(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_21(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_14(x)*F_3(x))
Eq(F_21(x), F_17(x)*F_3(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_13(x)*F_24(x)*F_3(x))
Eq(F_24(x), F_13(x) + F_25(x) + F_26(x))
Eq(F_25(x), F_24(x)*F_3(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_13(x)*F_24(x)*F_3(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_13(x)*F_3(x)*F_7(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": [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], "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, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"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, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"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": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "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": [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_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, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"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, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"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, 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