0231_1032_3021
Counting sequence:
1, 1, 2, 6, 21, 75, 268, 961, 3467, 12591, 46012, 169088, 624478, 2316582, 8627816, 32247951, 120920851, 454745327, 1714709308, 6481410516, 24553812570, 93210010594, 354513607608, 1350734328226, 5154896687614, 19703112257158, 75417330272472, 289061147718768, 1109315227745564, 4262207197809132, 16394529350208080, 63127872150354299, 243319195059356035, 938732439659539503, 3624903992618113724, 14009447588729048556, 54187281583304283778, 209753347193002944970, 812531346865591361560, 3149753830411641728414, 12218134689518789930762, 47425557026857635397570, 184199075446281785570120, 715844844061172059995808, 2783533554374030443288548, 10829549802450228160992084, 42155241206283565113399472, 164176642089958409615738742, 639707794702241600588870062, 2493760382017300751561079622, 9725755433416270100997406744, 37947279086867833888283949032, 148122182164807121048756683460, 578409677713069951341686642580, 2259545692150538661437288591920, 8830214716857140334584237094404, 34520784985600100090959920728476, 135003268460334851433156315854508, 528150046475522138138168805961776, 2066879712138009535379817159108256, 8091211520498013219119330806824888, 31684598482185109712678426589353560, 124112400806854324633785934262948512, 486307143789439696800876809799030739, 1906035474699076133294323989229216739, 7472616161392662147746227190185030319, 29304370072303662860088676085107141308, 114949455393500279565703277852265455452, 451017279887673190901148417042885760274, 1770060974057167515023304285885191624410, 6948470503973018515122020876789082389976, 27283082092013477630983190880362353646022, 107151496480513889679987758480472213992914, 420921520409820660702850929191900700481962, 1653863537447159581217275576890540786514408, 6499673340513565578186878498517854717891008, 25549032201868076778622173651087697884194452, 100449151494499671736669703989056292773601220, 395007093800775719659404643521154229658538736, 1553632518995758681199368221932804141095113954, 6111876189743825837070221927461854128190834746, 24048154289250947003352190462522000102206933378, 94638552856394815644117092678816107230724050696, 372504750503336468078956831463137314208491936536, 1466463394737935847889955763054485702280081891660, 5774105072655307092943245154092934739318199347516, 22738959844399373455589915006135625587267024168080, 89562751243421643516166469332485962406767573690332, 352820400099364441313165259789919216209323234977316, 1390106069122815709237377892557820154139118882213780, 5477832940939355017108720603620581349583069309361360, 21589116308127645293545438651902647234201830282637984, 85099081990209356357737321537142276223496778136228744, 335488422672976343844022957515403220869173195409613032, 1322792180436330376611343567776095597357006897709025376, 5216340353810774273560837455282475234052373335285304926, 20573078276108688057840492886824500768502160164354283726, 81150381102103412771456379669154427000938436416187439302, 320139069800518431048134217790160837959486905175707965208, 1263114024845715942493336909906254724903723241196952320696, 4984265444874573487486532440775010779718849192734206869620
Generating function in Maple syntax:
((2*x^3-3*x^2+3*x-1)*(1-4*x)^(1/2)+4*x^4-14*x^3+17*x^2-7*x+1)/(8*x^4-16*x^3+12*x^2-2*x)
Generating function in latex syntax:
\frac{\left(2 x^{3}-3 x^{2}+3 x -1\right) \sqrt{1-4 x}+4 x^{4}-14 x^{3}+17 x^{2}-7 x +1}{8 x^{4}-16 x^{3}+12 x^{2}-2 x}
Generating function in sympy syntax:
(4*x**4 - 14*x**3 + 17*x**2 - 7*x + sqrt(1 - 4*x)*(2*x**3 - 3*x**2 + 3*x - 1) + 1)/(8*x**4 - 16*x**3 + 12*x**2 - 2*x)
Implicit equation for the generating function in Maple syntax:
x*(4*x^3-8*x^2+6*x-1)*F(x)^2+(-4*x^4+14*x^3-17*x^2+7*x-1)*F(x)+x^4-4*x^3+8*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(4 x^{3}-8 x^{2}+6 x -1\right) F \! \left(x \right)^{2}+\left(-4 x^{4}+14 x^{3}-17 x^{2}+7 x -1\right) F \! \left(x \right)+x^{4}-4 x^{3}+8 x^{2}-5 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 75
a(6) = 268
a(7) = 961
a(n+7) = 16*(2*n+1)/(n+8)*a(n)-24*(5*n+7)/(n+8)*a(1+n)+20*(11*n+28)/(n+8)*a(n+2)-48*(5*n+19)/(n+8)*a(n+3)+2*(82*n+407)/(n+8)*a(n+4)-(65*n+394)/(n+8)*a(n+5)+(92+13*n)/(n+8)*a(n+6), n >= 8
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) = 75
a \! \left(6\right) = 268
a \! \left(7\right) = 961
a \! \left(n +7\right) = \frac{16 \left(2 n +1\right) a \! \left(n \right)}{n +8}-\frac{24 \left(5 n +7\right) a \! \left(1+n \right)}{n +8}+\frac{20 \left(11 n +28\right) a \! \left(n +2\right)}{n +8}-\frac{48 \left(5 n +19\right) a \! \left(n +3\right)}{n +8}+\frac{2 \left(82 n +407\right) a \! \left(n +4\right)}{n +8}-\frac{\left(65 n +394\right) a \! \left(n +5\right)}{n +8}+\frac{\left(92+13 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 8
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/358/
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[4,x]
F[3,x] = x
F[4,x] = F[5,x]+F[8,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[5,x]^2*F[3,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[3,x]
F[10,x] = F[11,x]+F[21,x]+F[4,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[3,x]
F[13,x] = F[10,x]+F[14,x]+F[19,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[3,x]*F[4,x]*F[5,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[16,x]*F[3,x]
F[19,x] = F[20,x]
F[20,x] = F[13,x]*F[3,x]*F[5,x]
F[21,x] = F[22,x]
F[22,x] = F[16,x]*F[23,x]*F[26,x]*F[3,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[23,x]*F[3,x]
F[26,x] = F[27,x]+F[5,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[3,x]
F[29,x] = F[27,x]+F[30,x]
F[30,x] = F[26,x]*F[5,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_{4}\! \left(x \right)
F_{3}\! \left(x \right) = x
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)+F_{8}\! \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_{5} \left(x \right)^{2} F_{3}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{3}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{21}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{3}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{3}\! \left(x \right) F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right) F_{3}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right) F_{3}\! \left(x \right) F_{5}\! \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_{26}\! \left(x \right) F_{3}\! \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_{23}\! \left(x \right) F_{3}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{5}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{3}\! \left(x \right)
F_{29}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{26}\! \left(x \right) F_{5}\! \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)*F_4(x))
Eq(F_3(x), x)
Eq(F_4(x), F_5(x) + F_8(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_3(x)*F_5(x)**2)
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_3(x))
Eq(F_10(x), F_11(x) + F_21(x) + F_4(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_3(x))
Eq(F_13(x), F_10(x) + F_14(x) + F_19(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_3(x)*F_4(x)*F_5(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_16(x)*F_3(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_13(x)*F_3(x)*F_5(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_16(x)*F_23(x)*F_26(x)*F_3(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_23(x)*F_3(x))
Eq(F_26(x), F_27(x) + F_5(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_3(x))
Eq(F_29(x), F_27(x) + F_30(x))
Eq(F_30(x), F_26(x)*F_5(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "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.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "row_and_col_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": [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], "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": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [3, 0, 2, 1], "pos": [[1, 2], [1, 0], [1, 2], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}], "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": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 2, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"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": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [0, 1], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [3, 0, 2, 1], "pos": [[1, 2], [1, 0], [1, 2], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "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}}, {"children": [{"assumptions": [], "class_module": 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