0132_0231_0312_1023_1203_2013
Counting sequence:
1, 1, 2, 6, 18, 53, 163, 522, 1719, 5784, 19812, 68871, 242358, 861668, 3090432, 11167759, 40621301, 148605317, 546411737, 2018230169, 7484861486, 27860359774, 104047442153, 389753163855, 1464029290928, 5513328208975, 20811125079454, 78726035145018, 298410535581677, 1133240023960240, 4311087803426268, 16427053012612855, 62689613176817816, 239581949296097848, 916849869860793558, 3513130249342128680, 13477559855059595645, 51763176001625760290, 199020166428352050851, 765976949706316305906, 2950900089222760411963, 11378706211035180133843, 43914880600586060936595, 169626298146093504276448, 655723506839130035698915, 2536757527408501184927701, 9820946813848903228282052, 38047878324641781950261982, 147501877816206578153720808, 572193329573742206002379771, 2221032947133498397008424723, 8626282430748462192470626911, 33522693716701123550500338883, 130344098757300664585215598183, 507075061666098262525806923450, 1973663971426045235819791017920, 7685752120135989373908900062621, 29943617467683270317038518369643, 116713155608496464720230078364040, 455120388698776713057285653907077, 1775486301713990269574189998405282, 6929259855355835641567445921924666, 27053850559895829091538055286852023, 105666797619694464262290157605680859, 412866825121609812071041072251011648, 1613758034111202078042512900958666729, 6309850472949315790827659595132800478, 24680126717693427288991992992133338560, 96564838391277966342288462339350709601, 377946044011775757604618317076706884734, 1479707106467438688354522320299028001806, 5794995063166647613656908554206286771318, 22701684351081785014603899178444604364390, 88958466226156863809661260621728804206297, 348688259468992229131438626510776276052232, 1367114747015185089874210673653247604473311, 5361509954172824702794247932513918888392288, 21032010581049265587231911950104731168661467, 82524543160914045084475674811032965075424347, 323885401377550384447426677500763658036517237, 1271460502277087549262867694682050119602023194, 4992465541534366483881088759033953355090812869, 19607648888258880964335298024676144654620820777, 77025023452397987664142447921551238358596377028, 302643773693689965486419917947094420049331805984, 1189386567973916256749356333910468249367568366824, 4675236333863479721164788094343484644231400322097, 18381091994275607917196863359992539633852990968277, 72281018418075661263703656062099364001008827084559, 284289300765831056028908226445785557015780832321017, 1118351210837101041413254332644506172711553317122717, 4400231866723717390792170393718015718361894646564228, 17316129719923617454926380404279528172379425963803402, 68155724824281555119628353637273154797546023217733177, 268304816278017814284488916965567595052084142442074425, 1056398173335139685629687769884634797451701098594618162, 4160047947330366978302376149172381442151547997509418043, 16384721970467339121948942603536608702854907517604318884, 64542893470273969215910938461376137054101392543310454234, 254287500382722503738674534174483419927644702676112526565, 1001999432858673338775447088133107196113559329908439560538
Generating function in Maple syntax:
1/2*(-(x-1)^2*(-4*x+1)^(1/2)+2*x^5-2*x^3+x^2-2*x+1)/(x^7+2*x^6+x^5+x^4-2*x+1)/x
Generating function in latex syntax:
\frac{-\left(x -1\right)^{2} \sqrt{-4 x +1}+2 x^{5}-2 x^{3}+x^{2}-2 x +1}{2 \left(x^{7}+2 x^{6}+x^{5}+x^{4}-2 x +1\right) x}
Generating function in sympy syntax:
(x**5 - x**3 + x**2/2 - x - sqrt(1 - 4*x)*(x - 1)**2/2 + 1/2)/(x*(x**7 + 2*x**6 + x**5 + x**4 - 2*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^7+2*x^6+x^5+x^4-2*x+1)*x*F(x)^2-(x-1)*(2*x^4+2*x^3+x-1)*F(x)+(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{7}+2 x^{6}+x^{5}+x^{4}-2 x +1\right) x F \! \left(x \right)^{2}-\left(x -1\right) \left(2 x^{4}+2 x^{3}+x -1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 53
a(6) = 163
a(7) = 522
a(8) = 1719
a(n+9) = -2*(11+2*n)/(10+n)*a(n)-3*(1+n)/(10+n)*a(1+n)+5*a(n+2)-(1+n)/(10+n)*a(n+3)+(31+4*n)/(10+n)*a(n+4)-a(n+5)+4*(11+2*n)/(10+n)*a(n+6)-2*(52+7*n)/(10+n)*a(n+7)+(61+7*n)/(10+n)*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) = 18
a \! \left(5\right) = 53
a \! \left(6\right) = 163
a \! \left(7\right) = 522
a \! \left(8\right) = 1719
a \! \left(n +9\right) = -\frac{2 \left(11+2 n \right) a \! \left(n \right)}{10+n}-\frac{3 \left(1+n \right) a \! \left(1+n \right)}{10+n}+5 a \! \left(n +2\right)-\frac{\left(1+n \right) a \! \left(n +3\right)}{10+n}+\frac{\left(31+4 n \right) a \! \left(n +4\right)}{10+n}-a \! \left(n +5\right)+\frac{4 \left(11+2 n \right) a \! \left(n +6\right)}{10+n}-\frac{2 \left(52+7 n \right) a \! \left(n +7\right)}{10+n}+\frac{\left(61+7 n \right) a \! \left(n +8\right)}{10+n}, \quad n \geq 9
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://www.permpal.com/tree/22284/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[10,x]*F[3,x]
F[3,x] = F[1,x]+F[14,x]+F[4,x]
F[4,x] = F[10,x]*F[5,x]
F[5,x] = F[1,x]+F[11,x]+F[14,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[0,x]*F[10,x]*F[8,x]
F[8,x] = F[1,x]+F[9,x]
F[9,x] = F[10,x]*F[8,x]
F[10,x] = x
F[11,x] = F[10,x]*F[12,x]
F[12,x] = F[1,x]+F[11,x]+F[13,x]+F[14,x]
F[13,x] = F[0,x]*F[10,x]
F[14,x] = F[15,x,1]
F[15,x,k[0]] = F[10,x]*F[16,x,k[0]]
F[16,x,k[0]] = F[1,x]+F[17,x,k[0]]+F[22,x,k[0]]+F[4,x]
F[17,x,k[0]] = F[18,x,k[0]]*F[19,x,k[0]]
F[18,x,k[0]] = k[0]*x
F[19,x,k[0]] = (F[20,x,k[0]]*k[0]-F[20,x,1])/(-1+k[0])
F[20,x,k[0]] = F[1,x]+F[15,x,k[0]]+F[21,x,k[0]]
F[21,x,k[0]] = F[18,x,k[0]]*F[20,x,k[0]]
F[22,x,k[0]] = F[10,x]*F[23,x,k[0]]
F[23,x,k[0]] = (F[16,x,k[0]]*k[0]-F[16,x,1])/(-1+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_{10}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{10}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{0}\! \left(x \right) F_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{1}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{13}\! \left(x \right)+F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = F_{0}\! \left(x \right) F_{10}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x , 1\right)
F_{15}\! \left(x , y\right) = F_{10}\! \left(x \right) F_{16}\! \left(x , y\right)
F_{16}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{17}\! \left(x , y\right)+F_{22}\! \left(x , y\right)+F_{4}\! \left(x \right)
F_{17}\! \left(x , y\right) = F_{18}\! \left(x , y\right) F_{19}\! \left(x , y\right)
F_{18}\! \left(x , y\right) = y x
F_{19}\! \left(x , y\right) = \frac{y F_{20}\! \left(x , y\right)-F_{20}\! \left(x , 1\right)}{-1+y}
F_{20}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{21}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{18}\! \left(x , y\right) F_{20}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = F_{10}\! \left(x \right) F_{23}\! \left(x , y\right)
F_{23}\! \left(x , y\right) = \frac{y F_{16}\! \left(x , y\right)-F_{16}\! \left(x , 1\right)}{-1+y}
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_10(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_14(x) + F_4(x))
Eq(F_4(x), F_10(x)*F_5(x))
Eq(F_5(x), F_1(x) + F_11(x) + F_14(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_0(x)*F_10(x)*F_8(x))
Eq(F_8(x), F_1(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_8(x))
Eq(F_10(x), x)
Eq(F_11(x), F_10(x)*F_12(x))
Eq(F_12(x), F_1(x) + F_11(x) + F_13(x) + F_14(x))
Eq(F_13(x), F_0(x)*F_10(x))
Eq(F_14(x), F_15(x, 1))
Eq(F_15(x, k_0), F_10(x)*F_16(x, k_0))
Eq(F_16(x, k_0), F_1(x) + F_17(x, k_0) + F_22(x, k_0) + F_4(x))
Eq(F_17(x, k_0), F_18(x, k_0)*F_19(x, k_0))
Eq(F_18(x, k_0), k_0*x)
Eq(F_19(x, k_0), (-k_0*F_20(x, k_0) + F_20(x, 1))/(1 - k_0))
Eq(F_20(x, k_0), F_1(x) + F_15(x, k_0) + F_21(x, k_0))
Eq(F_21(x, k_0), F_18(x, k_0)*F_20(x, k_0))
Eq(F_22(x, k_0), F_10(x)*F_23(x, k_0))
Eq(F_23(x, k_0), (-k_0*F_16(x, k_0) + F_16(x, 1))/(1 - k_0))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "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.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": "row_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, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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], "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, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 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": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 1, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"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, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [1, 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