0132_0213_0231_1032_1320
Counting sequence:
1, 1, 2, 6, 19, 59, 185, 594, 1956, 6585, 22576, 78555, 276688, 984515, 3533442, 12775898, 46492539, 170150335, 625834061, 2312212827, 8577099703, 31932147389, 119273897655, 446854671751, 1678729949921, 6322552995849, 23867997792453, 90297502326105, 342297947424865, 1299993812652933, 4945753113403585, 18846415638138194, 71926004931904524, 274892996725823465, 1052023107458635680, 4031226670290871679, 15465671342751015364, 59400724543364608187, 228391608810317309078, 879042794467338273579, 3386563396533112329838, 13058917901804330013707, 50400507428525553014756, 194681481741342769800885, 752592503713442680323154, 2911556640788517507268883, 11272138474136691661240572, 43670647738251484624517791, 169302188483965918935818540, 656769959833003425717399111, 2549356793030797176363012646, 9901571190291468251517043109, 38479009695413522514068117616, 149616890355543389999047353247, 582057021983100301501401467222, 2265532514627285430198356936889, 8822406343391136582505989177232, 34372280920532054213411923008803, 133976035235393188218006256027422, 522440285387623899010599809617033, 2038123950432825066534314821891676, 7954316534917429464194691096600807, 31056157929521838965853959178062242, 121299686110459985864623618370491946, 473951051782566896929729638478247835, 1852525445657096091251804042262488623, 7243475382182745651792165984831222461, 28332008522705313590782693026790989699, 110853896201973176395351372201033501087, 433874000539660837037147482964274868925, 1698679236832367541812633442027099948063, 6652584634838602131412704905767104791251, 26061359394508280705146460153486572648589, 102124007421740396238720398625686061737577, 400294293854581266324531272804080906376305, 1569453647821039048986216336975788760945713, 6155056378139473365692407071280288349490381, 24144991007096316176727884737307124807524029, 94739408899273186584571288214514805112200757, 371826339489099221234453797004268107170733115, 1459663808693083939052417907472725859779651663, 5731472155985852884867086896122453230434889981, 22510116053310271709930589920236017131847878359, 88427045173704460469547028917605774269685411697, 347444979285073436680109221246572061415812094575, 1365457894177774489796647837331008904994178163993, 5367348610930891712841979264323045476143491274971, 21102235287949427398526008092441921406349882622929, 82981695048353473595068906933860138114887674331667, 326376897839443024617028855763552749220781215039705, 1283919912210454500268076266786325168426642119200823, 5051682922232640911624617834229355948882889311930341, 19879805236970358855810140423708521881389669606574059, 78246403232573308022770844558548727692803138180697977, 308028736855588966570892141917784363406061837567825495, 1212805362047599190865497623455209910522328601049860663, 4775979801220739419205377126455125095288529934002332481, 18810653285176226501483489617524840231233510417217420249, 74099259874152397959073766069511011473451222393380477717, 291938296240438894630406745545244640895395943790185605025, 1150360949545285331418936309784661257971541682401238493513
Generating function in Maple syntax:
1/2*((-2*x^3+x^2+x-1)*(1-4*x)^(1/2)-2*x^4+4*x^3-3*x^2-x+1)/x/(x-1)^2
Generating function in latex syntax:
\frac{\left(-2 x^{3}+x^{2}+x -1\right) \sqrt{1-4 x}-2 x^{4}+4 x^{3}-3 x^{2}-x +1}{2 x \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**4 + 2*x**3 - 3*x**2/2 - x/2 + sqrt(1 - 4*x)*(-2*x**3 + x**2 + x - 1)/2 + 1/2)/(x*(x - 1)**2)
Implicit equation for the generating function in Maple syntax:
x*(x-1)^4*F(x)^2+(2*x^4-4*x^3+3*x^2+x-1)*(x-1)^2*F(x)+x^7+2*x^5-7*x^4+6*x^3+x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{4} F \! \left(x \right)^{2}+\left(2 x^{4}-4 x^{3}+3 x^{2}+x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{7}+2 x^{5}-7 x^{4}+6 x^{3}+x^{2}-3 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 59
a(6) = 185
a(7) = 594
a(n+5) = -4*(-1+2*n)/(6+n)*a(n)+2*(-2+7*n)/(6+n)*a(1+n)-3*(n-3)/(6+n)*a(n+2)-(35+8*n)/(6+n)*a(n+3)+(29+6*n)/(6+n)*a(n+4)-1/(6+n), 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) = 19
a \! \left(5\right) = 59
a \! \left(6\right) = 185
a \! \left(7\right) = 594
a \! \left(n +5\right) = -\frac{4 \left(-1+2 n \right) a \! \left(n \right)}{6+n}+\frac{2 \left(-2+7 n \right) a \! \left(1+n \right)}{6+n}-\frac{3 \left(n -3\right) a \! \left(n +2\right)}{6+n}-\frac{\left(35+8 n \right) a \! \left(n +3\right)}{6+n}+\frac{\left(29+6 n \right) a \! \left(n +4\right)}{6+n}-\frac{1}{6+n}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21562/
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[13,x]
F[12,x] = F[0,x]*F[7,x]
F[13,x] = F[14,x]+F[18,x]
F[14,x] = F[15,x]*F[7,x]
F[15,x] = F[16,x]
F[16,x] = F[10,x]*F[17,x]
F[17,x] = F[1,x]+F[15,x]
F[18,x] = F[19,x]
F[19,x] = F[10,x]*F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[10,x]*F[22,x]
F[22,x] = F[17,x]+F[23,x]
F[23,x] = F[20,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_{13}\! \left(x \right)
F_{12}\! \left(x \right) = F_{0}\! \left(x \right) F_{7}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{7}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{17}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{20}\! \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_13(x))
Eq(F_12(x), F_0(x)*F_7(x))
Eq(F_13(x), F_14(x) + F_18(x))
Eq(F_14(x), F_15(x)*F_7(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_10(x)*F_17(x))
Eq(F_17(x), F_1(x) + F_15(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_10(x)*F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_10(x)*F_22(x))
Eq(F_22(x), F_17(x) + F_23(x))
Eq(F_23(x), F_20(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, 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, 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, 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, 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, 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, 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"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"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], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 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, 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], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 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, 3, 2], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[2, 0], [2, 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}}, {"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, 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, 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": []}, {"assumptions": [], 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