0231_1302_2130

Counting sequence:
1, 1, 2, 6, 21, 77, 286, 1069, 4018, 15182, 57636, 219701, 840422, 3224664, 12405795, 47838633, 184854955, 715622717, 2774949566, 10776389502, 41906150394, 163160782026, 635976849603, 2481498550141, 9691665102851, 37884622072298, 148210754267149, 580261383608007, 2273381849797675, 8912626064141355, 34962728878629444, 137231802226244892, 538938226156154232, 2117601726394445729, 8324499212431746975, 32739270878290130200, 128814984354241899256, 507038033968148260125, 1996560545042446097351, 7864731868227479573284, 30991111271959570837911, 122161687104696526716630, 481693613446490874201997, 1899934158673633388123330, 7496045688584238088042014, 29583292098898294971666631, 116782109835633755859843673, 461123146970850441947249127, 1821226080774235939790350350, 7194704136089691992948063486, 28428910251441760066700378579, 112357472545216787349488506609, 444155124022329133071026513129, 1756123456887320936090687260967, 6944802953908665415772243074716, 27469222129010165668445829694369, 108670466611560868175515708725437, 429984472110266925372374437576034, 1701639469179902452055124809547629, 6735245953659951170803515243930322, 26662950812431439634339934118882042, 105567351837461504321799732742028040, 418037875524562457149322986987895260, 1655633657412233663863068804608446313, 6558032783204274817794968558965728793, 25980161090722373498608538965348214074, 102936001174619640516636295547845880284, 407894919750639427578457147419587374846, 1616528158057200205367855823106299533184, 6407235943338350871315936100209515451830, 25398564995207080843463298786291021505719, 100692535551619521788343936386050985936783, 399239661816212799071108572590022252452786, 1583131937211407085529057235364941736435542, 6278361892208717864277753347983924276630255, 24901196032296284327986638148722325419728163, 98772845245049298661556718735146107922588428, 391829678339231252433334308837136533813940423, 1554527645829310020637223281142106165827169118, 6167936905709762399191913816263837127799789303, 24474891407674258446840184178556514525406960004, 97127020580220846260897618458856905109367304548, 385475605332398438690727633962906302017932985263, 1529996367346332178717244501476042839679384671893, 6073229989642469758975471971528252370330600073063, 24109270645423667638852965066875581315162340143094, 95715583003841167448126926412852516345753997754638, 380027216417947739010431641636291567532217019808349, 1508966141408403768580545675982464117878369669891953, 5992062358849910138906961569939474297002308100461875, 23796029774218532707194523080209837010245172306939131, 94506867870552325995835458347446056235956061124149574, 375363709250352811516721950961594738905780321607923025, 1490975881059496613222216833811119689998303312707261415, 5922673286459272417983952673937204811640115705571210780, 23528442157585731809041653574744584425549444613664267459, 93475165539304261776997530196972739362125074205071001594, 371386777474984310835744815846095446628773368925824728102, 1475649532247980156126113277778173754890247361565012503700, 5863623649948511010129320573408311471426120521598160622180, 23300998197256942249384249671973668093899205864895577481909

Generating function in Maple syntax:
(-4*(x^2-x+1)*(x-1)^2*(x-1/2)^2*(1-4*x)^(1/2)-2*x^7+18*x^6-22*x^5-7*x^4+35*x^3-28*x^2+9*x-1)/(2*x^7-16*x^6+76*x^5-146*x^4+144*x^3-76*x^2+20*x-2)

Generating function in latex syntax:
\frac{-4 \left(x^{2}-x +1\right) \left(x -1\right)^{2} \left(x -\frac{1}{2}\right)^{2} \sqrt{1-4 x}-2 x^{7}+18 x^{6}-22 x^{5}-7 x^{4}+35 x^{3}-28 x^{2}+9 x -1}{2 x^{7}-16 x^{6}+76 x^{5}-146 x^{4}+144 x^{3}-76 x^{2}+20 x -2}

Generating function in sympy syntax:
(-2*x**7 + 18*x**6 - 22*x**5 - 7*x**4 + 35*x**3 - 28*x**2 + 9*x + sqrt(1 - 4*x)*(x - 1)**2*(x - 1/2)**2*(-4*x**2 + 4*x - 4) - 1)/(2*x**7 - 16*x**6 + 76*x**5 - 146*x**4 + 144*x**3 - 76*x**2 + 20*x - 2)

Implicit equation for the generating function in Maple syntax:
(x^7-8*x^6+38*x^5-73*x^4+72*x^3-38*x^2+10*x-1)*F(x)^2+(2*x^7-18*x^6+22*x^5+7*x^4-35*x^3+28*x^2-9*x+1)*F(x)+x^2*(x^5+6*x^4-19*x^3+22*x^2-11*x+2) = 0

Implicit equation for the generating function in latex syntax:
\left(x^{7}-8 x^{6}+38 x^{5}-73 x^{4}+72 x^{3}-38 x^{2}+10 x -1\right) F \! \left(x \right)^{2}+\left(2 x^{7}-18 x^{6}+22 x^{5}+7 x^{4}-35 x^{3}+28 x^{2}-9 x +1\right) F \! \left(x \right)+x^{2} \left(x^{5}+6 x^{4}-19 x^{3}+22 x^{2}-11 x +2\right) = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 77
a(6) = 286
a(7) = 1069
a(8) = 4018
a(9) = 15182
a(10) = 57636
a(11) = 219701
a(12) = 840422
a(n+12) = -4*(1+2*n)/(n+12)*a(n)+2*(43*n+49)/(n+12)*a(n+1)-(831+509*n)/(n+12)*a(n+2)+3*(558*n+1391)/(n+12)*a(n+3)-3*(1156*n+4081)/(n+12)*a(n+4)+(4906*n+22841)/(n+12)*a(n+5)-(28563+4922*n)/(n+12)*a(n+6)+13*(272*n+1885)/(n+12)*a(n+7)-(14410+1797*n)/(n+12)*a(n+8)+(625*n+5667)/(n+12)*a(n+9)-(1411+140*n)/(n+12)*a(n+10)+(18*n+199)/(n+12)*a(n+11), n >= 13

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) = 21
a \! \left(5\right) = 77
a \! \left(6\right) = 286
a \! \left(7\right) = 1069
a \! \left(8\right) = 4018
a \! \left(9\right) = 15182
a \! \left(10\right) = 57636
a \! \left(11\right) = 219701
a \! \left(12\right) = 840422
a \! \left(n +12\right) = -\frac{4 \left(1+2 n \right) a \! \left(n \right)}{n +12}+\frac{2 \left(43 n +49\right) a \! \left(n +1\right)}{n +12}-\frac{\left(831+509 n \right) a \! \left(n +2\right)}{n +12}+\frac{3 \left(558 n +1391\right) a \! \left(n +3\right)}{n +12}-\frac{3 \left(1156 n +4081\right) a \! \left(n +4\right)}{n +12}+\frac{\left(4906 n +22841\right) a \! \left(n +5\right)}{n +12}-\frac{\left(28563+4922 n \right) a \! \left(n +6\right)}{n +12}+\frac{13 \left(272 n +1885\right) a \! \left(n +7\right)}{n +12}-\frac{\left(14410+1797 n \right) a \! \left(n +8\right)}{n +12}+\frac{\left(625 n +5667\right) a \! \left(n +9\right)}{n +12}-\frac{\left(1411+140 n \right) a \! \left(n +10\right)}{n +12}+\frac{\left(18 n +199\right) a \! \left(n +11\right)}{n +12}, \quad n \geq 13

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/372/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[0,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[4,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[10,x]*F[5,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]^2*F[4,x]
F[13,x] = F[14,x]
F[14,x] = F[15,x]*F[17,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[15,x]*F[4,x]
F[17,x] = F[18,x]+F[24,x]
F[18,x] = F[19,x]*F[2,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[19,x]*F[22,x]*F[4,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[22,x]*F[4,x]
F[24,x] = F[11,x]*F[2,x]*F[22,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{0}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \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_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10} \left(x \right)^{2} F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{24}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{2}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{19}\! \left(x \right) F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right) F_{2}\! \left(x \right) F_{22}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_0(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_4(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_5(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)**2*F_4(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_15(x)*F_17(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_15(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_24(x))
Eq(F_18(x), F_19(x)*F_2(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_19(x)*F_22(x)*F_4(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_22(x)*F_4(x))
Eq(F_24(x), F_11(x)*F_2(x)*F_22(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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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, 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": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 1], [0, 1], [0, 1], [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": [1, 2, 0], "pos": [[1, 0], [1, 2], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 2], [1, 2], [1, 2], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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]]}], 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