0231_1032_1302_3021
Counting sequence:
1, 1, 2, 6, 20, 66, 216, 711, 2370, 8013, 27467, 95347, 334745, 1187040, 4246525, 15309103, 55565653, 202887665, 744734887, 2746588710, 10172272248, 37817461373, 141078288287, 527941804527, 1981309468104, 7455144430367, 28119447507012, 106297842502058, 402659493628703, 1528212587233699, 5810386389607075, 22128421087781109, 84406077557240152, 322428126416721422, 1233360004906250364, 4723998220848054280, 18115931217576145385, 69552615090182671135, 267326042926249993189, 1028535607528749088494, 3961181926451178715045, 15269910274914444144268, 58916209863880331563942, 227510532071107640145643, 879264695863028946772125, 3400739543277260154084997, 13162789660172265264523666, 50983486396644269160000027, 197608307417741023470533636, 766412902206586322821228254, 2974342366316930594475721059, 11549918685778566218839613331, 44876256354336077706259940605, 174459458587692505362051671474, 678584044970995941565974295167, 2640799956543332702817982234828, 10282105892406452218913301766336, 40053064234981696644617771465387, 156095119568385940721259233851875, 608605637335304724252932407420305, 2373937035986370698828054932040726, 9263664822577576985271891041094683, 36163546622368298727864001376609528, 141230390596410014944420732690337753, 551758389934763780379219610397006898, 2156394917623973505595554847054784261, 8430656247073117361772275994628060163, 32971869342692473545897184839419790735, 128994278847278558436068453914941192397, 504821444281311201979059726583735287648, 1976248567088560049100362780560888567413, 7738876206309328432040607981594980605428, 30313988640018085306953153840075886804124, 118777332350750561720831967896977364485255, 465528144907681704696033761892453052846005, 1825059130303121802372720983699100613125599, 7156874767123746723408655092779740213225034, 28072582037916773294753444183208256984342345, 110141469016153537828091963308901170471655196, 432241182843467606653400801560471593867344190, 1696701760306905917736528148566239640537752439, 6661720117295051184527181430830092463994057709, 26161718593060581661137775202216420087536000791, 102764419635169793695827171602942851827029998096, 403750972941558971362702962997613267953854152845, 1586632928835143040610966009426136810675674257968, 6236332554454194637358548982425256891192404162970, 24517145942510851703812577304495497726456255480971, 96404297811747700891356781987869805309359104877907, 379146261254127684502926301856931849345975313225063, 1491417329787505419911984519398660043365440381204484, 5867752759678770510812313188495296269209143439667149, 23089944521794952003713847225447573276478563254037584, 90876321464563213083742464492141884509888641478174414, 357728642621444716301144152787262566567497209140131595, 1408413655388602958917123832450558702511936368540134665, 5545985602671823761363778731041936660179280545319226905, 21842269210374991316134040139527665904370569276388711704, 86037114213455659549421647590423228160326377963454608485, 338954633860621580914010229727681617093508690078560371630, 1335560658501694324647815965527603803119285743574994219366
Generating function in Maple syntax:
(-3*(x^2-x+1/3)*(x-1)^3*(1-4*x)^(1/2)+4*x^6-5*x^5+11*x^3-13*x^2+6*x-1)/(16*x^6-44*x^5+58*x^4-40*x^3+14*x^2-2*x)
Generating function in latex syntax:
\frac{-3 \left(x^{2}-x +\frac{1}{3}\right) \left(x -1\right)^{3} \sqrt{1-4 x}+4 x^{6}-5 x^{5}+11 x^{3}-13 x^{2}+6 x -1}{16 x^{6}-44 x^{5}+58 x^{4}-40 x^{3}+14 x^{2}-2 x}
Generating function in sympy syntax:
(4*x**6 - 5*x**5 + 11*x**3 - 13*x**2 + 6*x + sqrt(1 - 4*x)*(x - 1)**3*(-3*x**2 + 3*x - 1) - 1)/(16*x**6 - 44*x**5 + 58*x**4 - 40*x**3 + 14*x**2 - 2*x)
Implicit equation for the generating function in Maple syntax:
x*(4*x^4-9*x^3+10*x^2-5*x+1)*(2*x-1)^2*F(x)^2-(2*x-1)*(4*x^6-5*x^5+11*x^3-13*x^2+6*x-1)*F(x)+x^7+2*x^6-15*x^5+37*x^4-43*x^3+26*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(4 x^{4}-9 x^{3}+10 x^{2}-5 x +1\right) \left(2 x -1\right)^{2} F \! \left(x \right)^{2}-\left(2 x -1\right) \left(4 x^{6}-5 x^{5}+11 x^{3}-13 x^{2}+6 x -1\right) F \! \left(x \right)+x^{7}+2 x^{6}-15 x^{5}+37 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 66
a(6) = 216
a(7) = 711
a(8) = 2370
a(9) = 8013
a(n+9) = 48*(1+2*n)/(n+10)*a(n)-12*(71+40*n)/(n+10)*a(1+n)+2*(1581+559*n)/(n+10)*a(n+2)-(6014+1571*n)/(n+10)*a(n+3)+6*(1164+241*n)/(n+10)*a(n+4)-(5232+895*n)/(n+10)*a(n+5)+2*(1271+185*n)/(n+10)*a(n+6)-2*(387+49*n)/(n+10)*a(n+7)+(134+15*n)/(n+10)*a(n+8), n >= 10
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) = 66
a \! \left(6\right) = 216
a \! \left(7\right) = 711
a \! \left(8\right) = 2370
a \! \left(9\right) = 8013
a \! \left(n +9\right) = \frac{48 \left(1+2 n \right) a \! \left(n \right)}{n +10}-\frac{12 \left(71+40 n \right) a \! \left(1+n \right)}{n +10}+\frac{2 \left(1581+559 n \right) a \! \left(n +2\right)}{n +10}-\frac{\left(6014+1571 n \right) a \! \left(n +3\right)}{n +10}+\frac{6 \left(1164+241 n \right) a \! \left(n +4\right)}{n +10}-\frac{\left(5232+895 n \right) a \! \left(n +5\right)}{n +10}+\frac{2 \left(1271+185 n \right) a \! \left(n +6\right)}{n +10}-\frac{2 \left(387+49 n \right) a \! \left(n +7\right)}{n +10}+\frac{\left(134+15 n \right) a \! \left(n +8\right)}{n +10}, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21444/
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[12,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[12,x]*F[7,x]
F[7,x] = F[13,x]+F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[9,x]^2*F[12,x]
F[12,x] = x
F[13,x] = F[14,x]
F[14,x] = F[15,x]*F[24,x]
F[15,x] = F[16,x]+F[22,x]
F[16,x] = F[17,x]*F[19,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[12,x]*F[17,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[12,x]*F[17,x]*F[19,x]
F[22,x] = F[2,x]*F[23,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[23,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_{12}\! \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_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{12}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{13}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \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_{9} \left(x \right)^{2} F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{24}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{22}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{19}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \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_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_12(x)*F_7(x))
Eq(F_7(x), F_13(x) + F_8(x))
Eq(F_8(x), F_4(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_9(x)**2)
Eq(F_12(x), x)
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_15(x)*F_24(x))
Eq(F_15(x), F_16(x) + F_22(x))
Eq(F_16(x), F_17(x)*F_19(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_12(x)*F_17(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_12(x)*F_17(x)*F_19(x))
Eq(F_22(x), F_2(x)*F_23(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_23(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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": []}, {"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]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [2, 0]], [[1, 1]]], "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, 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