0132_0231_1032_1320
Counting sequence:
1, 1, 2, 6, 20, 67, 225, 762, 2609, 9034, 31618, 111752, 398491, 1432211, 5183651, 18878250, 69133001, 254418074, 940430452, 3490036692, 12998483380, 48570406942, 182029922138, 684064524276, 2577146144510, 9731601868032, 36826295878550, 139634713891692, 530433131049457, 2018440883612403, 7693094952264951, 29365805670859306, 112253177071150945, 429669891972461778, 1646711671900351684, 6318531378661203540, 24271841697142726816, 93336517328204504794, 359284921653421559662, 1384340665928997034124, 5338794540356120482158, 20607286069472930912956, 79608270789188841451492, 307779022520467650466464, 1190825172498575308277238, 4610742721774822135210606, 17864641469019677236173342, 69263706533437674446482244, 268715924727385194170765194, 1043148848388065281387819204, 4051855644907939610409908968, 15747319896171587744275977672, 61234347704470380181504030536, 238237746425215898009000739628, 927352454246280743260647553452, 3611520172448515393122985887032, 14071449307388341977190849504432, 54850951438861360463462871961420, 213904238016288592449923134318318, 834522740406626301810501383562228, 3257125618846897146498129898597053, 12717526859097108444859432258113331, 49674921406243210351453148278658847, 194103285927865761755733931388206634, 758725714963225983732594898074603409, 2966799557156509444191559933987803970, 11604824419257067597911022727545308484, 45407917935015979598399642842392774036, 177730972805679261728206525536638716968, 695871750212336143189743756878425644898, 2725373728527797565162340756674176761110, 10677020212863460764618180689159963072828, 41840582695261797706765914906995444206582, 164008253104218267951089951812623275980156, 643058961156127914237642000562575340010716, 2522028090235615912259579757178644061283280, 9893731436797736225275034459202901711492986, 38822056941781827892174416730260756278833594, 152371048480987233420604975103968482786490362, 598176328891746108611032647010634917707364332, 2348855530002477736517272551077442303436984910, 9225316156636468602790559868913170224134365772, 36241129554382666579021217108786657650630315288, 142401784884497085824698963715605486190278351064, 559654743534762976041217600514588746068769814584, 2199955408276401364191162749541820251817184263460, 8649568781785125768342528498973056705746051260364, 34014175742014664685048825841365438762183704182296, 133785317983116360083788623128402206109238737761588, 526305762401030947611788956563316379459281579178608, 2070843017866357770939740144449668752449079120376220, 8149554899638962812866313541582309106342675225355928, 32077210279312051368359528393931284639170437332369642, 126279736363286990617355920910954074339717618993758126, 497214232100129138979738970775718839297591254834147126, 1958054291561337909472688368287191158515015843576230980, 7712154691968569526234431020942224339628554616336764826, 30380515261099350552647502476845961740566818582509789204, 119696529366427726783345086696083430289412370021987684072, 471665011342700535774553267696691521201977750928011236232, 1858874818480059799670368788358637070566614485023988964960
Generating function in Maple syntax:
1/4*((x^4-2*x^3+3*x^2-3*x+1)*(1-4*x)^(1/2)+2*x^5-7*x^4+14*x^3-11*x^2+5*x-1)/(x^2-2*x+2)/(x-1/2)/x^2
Generating function in latex syntax:
\frac{\left(x^{4}-2 x^{3}+3 x^{2}-3 x +1\right) \sqrt{1-4 x}+2 x^{5}-7 x^{4}+14 x^{3}-11 x^{2}+5 x -1}{4 \left(x^{2}-2 x +2\right) \left(x -\frac{1}{2}\right) x^{2}}
Generating function in sympy syntax:
(x**5/2 - 7*x**4/4 + 7*x**3/2 - 11*x**2/4 + 5*x/4 + sqrt(1 - 4*x)*(x**4 - 2*x**3 + 3*x**2 - 3*x + 1)/4 - 1/4)/(x**2*(x - 1/2)*(x**2 - 2*x + 2))
Implicit equation for the generating function in Maple syntax:
x^2*(x^2-2*x+2)*(2*x-1)^2*F(x)^2-(2*x-1)*(2*x^5-7*x^4+14*x^3-11*x^2+5*x-1)*F(x)+x^6-4*x^5+12*x^4-17*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
x^{2} \left(x^{2}-2 x +2\right) \left(2 x -1\right)^{2} F \! \left(x \right)^{2}-\left(2 x -1\right) \left(2 x^{5}-7 x^{4}+14 x^{3}-11 x^{2}+5 x -1\right) F \! \left(x \right)+x^{6}-4 x^{5}+12 x^{4}-17 x^{3}+14 x^{2}-6 x +1 = 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) = 225
a(7) = 762
a(8) = 2609
a(n+8) = -2*(1+2*n)/(10+n)*a(n)-1/2*(246+97*n)/(10+n)*a(2+n)+(19*n+27)/(10+n)*a(n+1)+(81*n+319)/(10+n)*a(n+3)-1/2*(1012+191*n)/(10+n)*a(n+4)+5/2*(31*n+202)/(10+n)*a(n+5)-1/2*(590+77*n)/(10+n)*a(n+6)+2*(5*n+44)/(10+n)*a(n+7), 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) = 225
a \! \left(7\right) = 762
a \! \left(8\right) = 2609
a \! \left(n +8\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{10+n}-\frac{\left(246+97 n \right) a \! \left(2+n \right)}{2 \left(10+n \right)}+\frac{\left(19 n +27\right) a \! \left(n +1\right)}{10+n}+\frac{\left(81 n +319\right) a \! \left(n +3\right)}{10+n}-\frac{\left(1012+191 n \right) a \! \left(n +4\right)}{2 \left(10+n \right)}+\frac{5 \left(31 n +202\right) a \! \left(n +5\right)}{2 \left(10+n \right)}-\frac{\left(590+77 n \right) a \! \left(n +6\right)}{2 \left(10+n \right)}+\frac{2 \left(5 n +44\right) a \! \left(n +7\right)}{10+n}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21224/
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[10,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[10,x]*F[11,x]*F[7,x]
F[7,x] = F[1,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[7,x]^2*F[10,x]
F[10,x] = x
F[11,x] = F[12,x]+F[4,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[14,x]
F[14,x] = F[15,x]+F[21,x]
F[15,x] = F[16,x]*F[18,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[10,x]*F[16,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[10,x]*F[16,x]*F[18,x]
F[21,x] = F[2,x]*F[22,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[10,x]*F[22,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_{10}\! \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_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{7} \left(x \right)^{2} F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{4}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{18}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{2}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right) F_{22}\! \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_10(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_10(x)*F_11(x)*F_7(x))
Eq(F_7(x), F_1(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_7(x)**2)
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_4(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_14(x))
Eq(F_14(x), F_15(x) + F_21(x))
Eq(F_15(x), F_16(x)*F_18(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_10(x)*F_16(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_10(x)*F_16(x)*F_18(x))
Eq(F_21(x), F_2(x)*F_22(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_10(x)*F_22(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, 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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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, 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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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"}}, {"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, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 1], [0, 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": [[1, 2]]}, {"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, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 2], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 2], [0, 2], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [0, 2]], [[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, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, "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, 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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": 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