0123_0132_0213_0231_1023_2013
Counting sequence:
1, 1, 2, 6, 18, 56, 182, 607, 2064, 7132, 24970, 88383, 315748, 1137014, 4122762, 15039631, 55157790, 203255438, 752190764, 2794352648, 10417047964, 38956725596, 146108755556, 549442692378, 2071236137154, 7825588757910, 29628705425026, 112396368769481, 427147763868952, 1626066393981642, 6199914669487490, 23674337971087975, 90526536516107862, 346612747381020150, 1328774743002835196, 5099959508876572332, 19595815096854376396, 75373039823206303248, 290202798262564327380, 1118403537483985163730, 4314067214053083095848, 16655136778953867015872, 64352465118296619170980, 248840917725264008822894, 962948414306490775852760, 3729021125724632515829676, 14450541473381558896204932, 56034891086154917675625302, 217423512151639447120615564, 844145217501531134740558956, 3279293292119122813414244224, 12746358814067084639718868200, 49570772806058369059031651480, 192881378522442426895301417660, 750882241859914689665216688704, 2924573758637890232107476018232, 11396073290965109905020023839478, 44426578760887044768062514423002, 173268218797696654415926906262610, 676047336847038507989160104958333, 2638830605823213552137188208999168, 10304252014194394220224939444492786, 40251919447093507010963387336118834, 157295661399052508579302445407455031, 614896494260281029349045570026100710, 2404571715644615548954992915809640998, 9406312968804141557852044185206061164, 36808050237399466306344680673027720308, 144080036499614049437512472972569593004, 564155075779965990815870511373540109456, 2209648189945967240591377842076996395236, 8657133481298722088691529691712579310506, 33927198149305230143110608117420883710480, 132996884873894265714844288686082361593016, 521496451778740979632997407083968332179436, 2045382852960322116313796895171607549424658, 8024321015341849559550829033769292299457688, 31488327081093542636061105648995049632733876, 123593541618741970454498831524486764183937772, 485226207842875808510835431770812029837254338, 1905427764530665532690145359302304812773285572, 7484073964378578592100818005054458714782890084, 29402120824135368257088028157613032596149651080, 115534581653508724369488269758683628857746497520, 454083698159077891013011294786336052284999142088, 1785041440100811574192314369009753926012424589592, 7018545911734733504471483365766822061252094224600, 27601359891042595454607032420718838459572836199804, 108566644315299227230857710708214910432171685391300, 427113323860509378264753125856685407450755887190156, 1680617284392005949408500446865638107932533016554612, 6614116211929016001390128623092355600501392675058682, 26034572422154936319480106450681258367142866184547728, 102495100605650614086976429603578542645725050049042084, 403578700507076463613322825228875143252353511293782772, 1589367539308495596794375902737194909395391147306395830, 6260225124191488766785589132806011478169085914513800956, 24661736898212149183855552285554769229032359106034026428, 97168185860175078622814621609999551376253271036586705496, 382904017896387111128491702453825469529924072642941280408, 1509106383708943822391223134846672093410207988111974629432
Generating function in Maple syntax:
1/2*(-(x^2+1)^2*(-4*x+1)^(1/2)+3*x^4+4*x^2-2*x+1)/(x^3+2*x^2+2*x+2)/x^2
Generating function in latex syntax:
\frac{-\left(x^{2}+1\right)^{2} \sqrt{-4 x +1}+3 x^{4}+4 x^{2}-2 x +1}{2 \left(x^{3}+2 x^{2}+2 x +2\right) x^{2}}
Generating function in sympy syntax:
(3*x**4/2 + 2*x**2 - x - sqrt(1 - 4*x)*(x**2 + 1)**2/2 + 1/2)/(x**2*(x**3 + 2*x**2 + 2*x + 2))
Implicit equation for the generating function in Maple syntax:
(x^3+2*x^2+2*x+2)*x^2*F(x)^2+(-3*x^4-4*x^2+2*x-1)*F(x)+x^4+2*x^2-x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+2 x^{2}+2 x +2\right) x^{2} F \! \left(x \right)^{2}+\left(-3 x^{4}-4 x^{2}+2 x -1\right) F \! \left(x \right)+x^{4}+2 x^{2}-x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 56
a(n+6) = (1+2*n)/(8+n)*a(n)+(13+5*n)/(8+n)*a(2+n)+1/2*(2+7*n)/(8+n)*a(n+1)+1/2*(44+13*n)/(8+n)*a(n+3)+2*a(n+4)+3*(n+6)/(8+n)*a(n+5), n >= 6
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) = 56
a \! \left(n +6\right) = \frac{\left(1+2 n \right) a \! \left(n \right)}{8+n}+\frac{\left(13+5 n \right) a \! \left(2+n \right)}{8+n}+\frac{\left(2+7 n \right) a \! \left(n +1\right)}{16+2 n}+\frac{\left(44+13 n \right) a \! \left(n +3\right)}{16+2 n}+2 a \! \left(n +4\right)+\frac{3 \left(n +6\right) a \! \left(n +5\right)}{8+n}, \quad n \geq 6
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://www.permpal.com/tree/22047/
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[7,x]
F[3,x] = F[1,x]+F[22,x]+F[4,x]
F[4,x] = F[5,x]
F[5,x] = F[6,x]*F[7,x]*F[8,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = x
F[8,x] = F[9,x,1]
F[9,x,k[0]] = F[1,x]+F[10,x,k[0]]+F[12,x,k[0]]
F[10,x,k[0]] = F[11,x,k[0]]*F[9,x,k[0]]
F[11,x,k[0]] = k[0]*x
F[12,x,k[0]] = F[13,x,k[0]]*F[7,x]
F[13,x,k[0]] = F[1,x]+F[14,x,k[0]]+F[16,x,k[0]]+F[20,x,k[0]]
F[14,x,k[0]] = F[11,x,k[0]]*F[15,x,k[0]]
F[15,x,k[0]] = (F[9,x,k[0]]*k[0]-F[9,x,1])/(-1+k[0])
F[16,x,k[0]] = F[17,x,k[0]]
F[17,x,k[0]] = F[18,x,k[0]]*F[7,x]*F[8,x]
F[18,x,k[0]] = F[19,x,k[0]]+F[7,x]
F[19,x,k[0]] = F[1,x]+F[11,x,k[0]]
F[20,x,k[0]] = F[21,x,k[0]]*F[7,x]
F[21,x,k[0]] = (F[13,x,k[0]]*k[0]-F[13,x,1])/(-1+k[0])
F[22,x] = F[12,x,1]
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_{7}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right) F_{7}\! \left(x \right) F_{8}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = x
F_{8}\! \left(x \right) = F_{9}\! \left(x , 1\right)
F_{9}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = F_{11}\! \left(x , y\right) F_{9}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = y x
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{7}\! \left(x \right)
F_{13}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)+F_{16}\! \left(x , y\right)+F_{20}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = F_{11}\! \left(x , y\right) F_{15}\! \left(x , y\right)
F_{15}\! \left(x , y\right) = \frac{y F_{9}\! \left(x , y\right)-F_{9}\! \left(x , 1\right)}{-1+y}
F_{16}\! \left(x , y\right) = F_{17}\! \left(x , y\right)
F_{17}\! \left(x , y\right) = F_{18}\! \left(x , y\right) F_{7}\! \left(x \right) F_{8}\! \left(x \right)
F_{18}\! \left(x , y\right) = F_{19}\! \left(x , y\right)+F_{7}\! \left(x \right)
F_{19}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)
F_{20}\! \left(x , y\right) = F_{21}\! \left(x , y\right) F_{7}\! \left(x \right)
F_{21}\! \left(x , y\right) = \frac{y F_{13}\! \left(x , y\right)-F_{13}\! \left(x , 1\right)}{-1+y}
F_{22}\! \left(x \right) = F_{12}\! \left(x , 1\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_7(x))
Eq(F_3(x), F_1(x) + F_22(x) + F_4(x))
Eq(F_4(x), F_5(x))
Eq(F_5(x), F_6(x)*F_7(x)*F_8(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), x)
Eq(F_8(x), F_9(x, 1))
Eq(F_9(x, k_0), F_1(x) + F_10(x, k_0) + F_12(x, k_0))
Eq(F_10(x, k_0), F_11(x, k_0)*F_9(x, k_0))
Eq(F_11(x, k_0), k_0*x)
Eq(F_12(x, k_0), F_13(x, k_0)*F_7(x))
Eq(F_13(x, k_0), F_1(x) + F_14(x, k_0) + F_16(x, k_0) + F_20(x, k_0))
Eq(F_14(x, k_0), F_11(x, k_0)*F_15(x, k_0))
Eq(F_15(x, k_0), (-k_0*F_9(x, k_0) + F_9(x, 1))/(1 - k_0))
Eq(F_16(x, k_0), F_17(x, k_0))
Eq(F_17(x, k_0), F_18(x, k_0)*F_7(x)*F_8(x))
Eq(F_18(x, k_0), F_19(x, k_0) + F_7(x))
Eq(F_19(x, k_0), F_1(x) + F_11(x, k_0))
Eq(F_20(x, k_0), F_21(x, k_0)*F_7(x))
Eq(F_21(x, k_0), (-k_0*F_13(x, k_0) + F_13(x, 1))/(1 - k_0))
Eq(F_22(x), F_12(x, 1))
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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "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, 2, 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], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "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, 2, 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], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, 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