0132_0231_0312_1203
Counting sequence:
1, 1, 2, 6, 20, 67, 224, 753, 2557, 8782, 30493, 106940, 378423, 1349927, 4850432, 17541679, 63812042, 233360769, 857489624, 3164561325, 11724980295, 43598821174, 162655454542, 608661697567, 2283978375642, 8592573029690, 32403069209802, 122463259744391, 463783045773951, 1759756473247145, 6689043442180487, 25468269243627161, 97121098930829342, 370907208182556227, 1418459216794981335, 5431676236562758352, 20824986466494824050, 79935619557979474951, 307166568359661288540, 1181573662409007116772, 4549650535710850417826, 17534947044126471938871, 67642613353753534670959, 261160501047355927054187, 1009133799533296929514853, 3902368229820327832988746, 15101883370673071871300477, 58484857465594378637462538, 226648152554539011752756303, 878911451624894242909576815, 3410442742348549693756815912, 13241533726494780506644679800, 51441962225728338722845987537, 199958038058621668596169604848, 777666247618898093440602394770, 3026022207911759000124106804670, 11780598309804233756159541893245, 45885058416612808788960561044743, 178803766466815410410889558528689, 697070287263227057748807313081399, 2718719884577746936012922701210302, 10608014740490962187247522317595337, 41407556438803745482423318468137342, 161694515871484653421595050271106966, 631648992385172118398551338161371532, 2468402820445789477711168764855167879, 9649639381660809590494689321747939116, 37736037202384435425990852362981290686, 147620679497302380001403822706927886156, 577669645581747241527500573813469530753, 2261253521255973804648023642109413062889, 8854263082859724379396940893948500260365, 34680496359627005956931442598680893962709, 135876490001421272080079108819189539871796, 532507866859954283084842795891549862017499, 2087503106817977494069568958811666563264240, 8185485779037021018789791963164139543852697, 32105169525901125374080345611839056910791975, 125955092722431804130387155975771852469799368, 494269642862722242058582353466234965105273316, 1940067663679205386856998276124896764901045295, 7616789323886847178974859349236748037376185522, 29910703394724574899338645371581570518506786502, 117483940977685948404149570669478860216024723234, 461556904422591421600982876628906511502283394687, 1813696494949806877393553984450354859417211826459, 7128438737719611458393690145423382741186443194549, 28022867990037277803475345261626707408363107072515, 110183628180464716166567042238157434640165445889400, 433317203331351037432804325844577981413107928075459, 1704422584330749974399069745776056424786138626625392, 6705471007309325960504844640558765611925979497300887, 26385187967362014267504113444634571010754105688524733, 103840861304982900689273030268775547277420863591741074, 408744482531536838220587999520362000606038266227779336, 1609197896940379691207909436757835974821997838410169947, 6336353380107702946846593349760644146083517783369188464, 24954001792531708294937261821555255769847392287468607256, 98290249199761372772700076904995536635569357557853860806, 387211886746660014366677711314037624041284263106561297331, 1525645271244246732460279182952590512824377976000477054333
Generating function in Maple syntax:
-(3*(1-4*x)^(1/2)*x^2-3*(1-4*x)^(1/2)*x-x^2+(1-4*x)^(1/2)+3*x-1)*(2*x-1)*(x-1)/(18*x^5-40*x^4+36*x^3-14*x^2+2*x)
Generating function in latex syntax:
-\frac{\left(3 \sqrt{1-4 x}\, x^{2}-3 \sqrt{1-4 x}\, x -x^{2}+\sqrt{1-4 x}+3 x -1\right) \left(2 x -1\right) \left(x -1\right)}{18 x^{5}-40 x^{4}+36 x^{3}-14 x^{2}+2 x}
Generating function in sympy syntax:
(x - 1)*(2*x - 1)*(-3*x**2*sqrt(1 - 4*x) + x**2 + 3*x*sqrt(1 - 4*x) - 3*x - sqrt(1 - 4*x) + 1)/(18*x**5 - 40*x**4 + 36*x**3 - 14*x**2 + 2*x)
Implicit equation for the generating function in Maple syntax:
x*(9*x^4-20*x^3+18*x^2-7*x+1)*F(x)^2-(x-1)*(2*x-1)*(x^2-3*x+1)*F(x)+(x-1)^2*(2*x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
x \left(9 x^{4}-20 x^{3}+18 x^{2}-7 x +1\right) F \! \left(x \right)^{2}-\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+\left(x -1\right)^{2} \left(2 x -1\right)^{2} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 67
a(6) = 224
a(7) = 753
a(8) = 2557
a(n+9) = 108*(2*n+1)/(n+10)*a(n)-6*(179*n+283)/(n+10)*a(1+n)+3*(797*n+2088)/(n+10)*a(n+2)-2*(1559*n+5700)/(n+10)*a(n+3)+2*(1307*n+6150)/(n+10)*a(n+4)-4*(364*n+2099)/(n+10)*a(n+5)+(3668+537*n)/(n+10)*a(n+6)-14*(9*n+71)/(n+10)*a(n+7)+(152+17*n)/(n+10)*a(n+8), n >= 9
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) = 67
a \! \left(6\right) = 224
a \! \left(7\right) = 753
a \! \left(8\right) = 2557
a \! \left(n +9\right) = \frac{108 \left(2 n +1\right) a \! \left(n \right)}{n +10}-\frac{6 \left(179 n +283\right) a \! \left(1+n \right)}{n +10}+\frac{3 \left(797 n +2088\right) a \! \left(n +2\right)}{n +10}-\frac{2 \left(1559 n +5700\right) a \! \left(n +3\right)}{n +10}+\frac{2 \left(1307 n +6150\right) a \! \left(n +4\right)}{n +10}-\frac{4 \left(364 n +2099\right) a \! \left(n +5\right)}{n +10}+\frac{\left(3668+537 n \right) a \! \left(n +6\right)}{n +10}-\frac{14 \left(9 n +71\right) a \! \left(n +7\right)}{n +10}+\frac{\left(152+17 n \right) a \! \left(n +8\right)}{n +10}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/21205/
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[16,x]*F[4,x]
F[4,x] = F[43,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[16,x]*F[8,x]
F[8,x] = F[24,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[16,x]
F[12,x] = F[13,x]+F[17,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = x
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[16,x]*F[20,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]
F[22,x] = F[16,x]*F[23,x]
F[23,x] = F[14,x]+F[21,x]
F[24,x] = F[18,x]+F[25,x]
F[25,x] = F[26,x]+F[27,x]+F[29,x]
F[26,x] = 0
F[27,x] = F[16,x]*F[28,x]
F[28,x] = F[10,x]+F[25,x]
F[29,x] = F[16,x]*F[30,x]
F[30,x] = F[31,x]+F[36,x]
F[31,x] = F[18,x]+F[32,x]
F[32,x] = F[26,x]+F[33,x]+F[35,x]
F[33,x] = F[16,x]*F[34,x]
F[34,x] = F[14,x]+F[32,x]
F[35,x] = F[16,x]*F[31,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[38,x]
F[38,x] = F[16,x]*F[39,x]
F[39,x] = F[18,x]+F[37,x]
F[40,x] = F[41,x]
F[41,x] = F[16,x]*F[42,x]
F[42,x] = F[32,x]+F[40,x]
F[43,x] = F[44,x]
F[44,x] = F[16,x]*F[45,x]*F[5,x]
F[45,x] = F[4,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[16,x]*F[48,x]
F[48,x] = F[49,x,1]
F[49,x,k[0]] = (F[50,x,k[0]]*k[0]-F[50,x,1])/(-1+k[0])
F[50,x,k[0]] = F[51,x,k[0]]+F[55,x,k[0]]
F[51,x,k[0]] = F[0,x]+F[52,x,k[0]]
F[52,x,k[0]] = F[53,x,k[0]]
F[53,x,k[0]] = F[50,x,k[0]]*F[54,x,k[0]]
F[54,x,k[0]] = k[0]*x
F[55,x,k[0]] = F[56,x,k[0]]
F[56,x,k[0]] = F[16,x]*F[49,x,k[0]]
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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{43}\! \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_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{24}\! \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_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \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_{16}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{14}\! \left(x \right)+F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{18}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = 0
F_{27}\! \left(x \right) = F_{16}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{29}\! \left(x \right) = F_{16}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{36}\! \left(x \right)
F_{31}\! \left(x \right) = F_{18}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{26}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = F_{16}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{16}\! \left(x \right) F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{16}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{18}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{16}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{32}\! \left(x \right)+F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{16}\! \left(x \right) F_{45}\! \left(x \right) F_{5}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{16}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x , 1\right)
F_{49}\! \left(x , y\right) = \frac{y F_{50}\! \left(x , y\right)-F_{50}\! \left(x , 1\right)}{-1+y}
F_{50}\! \left(x , y\right) = F_{51}\! \left(x , y\right)+F_{55}\! \left(x , y\right)
F_{51}\! \left(x , y\right) = F_{0}\! \left(x \right)+F_{52}\! \left(x , y\right)
F_{52}\! \left(x , y\right) = F_{53}\! \left(x , y\right)
F_{53}\! \left(x , y\right) = F_{50}\! \left(x , y\right) F_{54}\! \left(x , y\right)
F_{54}\! \left(x , y\right) = y x
F_{55}\! \left(x , y\right) = F_{56}\! \left(x , y\right)
F_{56}\! \left(x , y\right) = F_{16}\! \left(x \right) F_{49}\! \left(x , y\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_16(x)*F_4(x))
Eq(F_4(x), F_43(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_16(x)*F_8(x))
Eq(F_8(x), F_24(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_16(x))
Eq(F_12(x), F_13(x) + F_17(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_16(x))
Eq(F_16(x), x)
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_16(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_16(x)*F_23(x))
Eq(F_23(x), F_14(x) + F_21(x))
Eq(F_24(x), F_18(x) + F_25(x))
Eq(F_25(x), F_26(x) + F_27(x) + F_29(x))
Eq(F_26(x), 0)
Eq(F_27(x), F_16(x)*F_28(x))
Eq(F_28(x), F_10(x) + F_25(x))
Eq(F_29(x), F_16(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_36(x))
Eq(F_31(x), F_18(x) + F_32(x))
Eq(F_32(x), F_26(x) + F_33(x) + F_35(x))
Eq(F_33(x), F_16(x)*F_34(x))
Eq(F_34(x), F_14(x) + F_32(x))
Eq(F_35(x), F_16(x)*F_31(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_16(x)*F_39(x))
Eq(F_39(x), F_18(x) + F_37(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_16(x)*F_42(x))
Eq(F_42(x), F_32(x) + F_40(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_16(x)*F_45(x)*F_5(x))
Eq(F_45(x), F_4(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_16(x)*F_48(x))
Eq(F_48(x), F_49(x, 1))
Eq(F_49(x, k_0), (-k_0*F_50(x, k_0) + F_50(x, 1))/(1 - k_0))
Eq(F_50(x, k_0), F_51(x, k_0) + F_55(x, k_0))
Eq(F_51(x, k_0), F_0(x) + F_52(x, k_0))
Eq(F_52(x, k_0), F_53(x, k_0))
Eq(F_53(x, k_0), F_50(x, k_0)*F_54(x, k_0))
Eq(F_54(x, k_0), k_0*x)
Eq(F_55(x, k_0), F_56(x, k_0))
Eq(F_56(x, k_0), F_16(x)*F_49(x, k_0))
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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "point_placements_tracked_fusion", "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": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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