0213_1302_2130
Counting sequence:
1, 1, 2, 6, 21, 76, 275, 989, 3539, 12631, 45066, 161021, 576887, 2074166, 7488003, 27150233, 98878251, 361680595, 1328574654, 4900021037, 18141052047, 67402330234, 251263851255, 939561899651, 3523414637736, 13248113693491, 49935804727105, 188651583360524, 714214447347319, 2709269984148373, 10296105377775928, 39195627933823077, 149450714483747458, 570703087995891436, 2182400046619221026, 8356683961339963456, 32038765176734800780, 122978513783712957414, 472570278511169215934, 1817864555599334526532, 6999884476976946280752, 26979416218954815840054, 104079758087783390596610, 401858585486681737608996, 1552873271444732894062198, 6005349818242114977305538, 23241544455554715144766496, 90012305359215505352875050, 348847628032008470827741324, 1352864351642388402736036056, 5249835414751561719832864796, 20384457190039743397159926936, 79196203352097499099401444712, 307858846719468809036670827556, 1197379582529076325603203244916, 4659468415401412986770404957896, 18140826154348768746292538686960, 70662045738579048443841840482724, 275369924283603598628553174286364, 1073596368778549050639858195962856, 4187484595895521014044848073656516, 16339803508224702064765627267611996, 63784612586541680621926319364683312, 249089174612301020898813888835047966, 973101735951180428918567966381927173, 3802953180722768285994759970177109628, 14867507279458968214170102927341912111, 58144029351916527277135565404960666455, 227466485248988910487809027250506699604, 890165536442637244132289452552228089207, 3484665752367367770831210358175044642217, 13645346530360481810306294917466948171844, 53448727103322631859340912737949982887565, 209418931485861275694906865913803398847701, 820761267540790348157340078447586778313218, 3217635277806828164588161593483184796422293, 12617481032765511230257649800270695013074841, 49490444276723333931794340045537630454970730, 194169034316783288519777253858593993325282869, 761983745693303922812865715595076783046234377, 2990996196393008984970356568643164762066022592, 11743237134900947536820806046352837354077465409, 46116794567549121154577768995720660112128182435, 181145393805211241651476143718074292269795977436, 711688668278590643152430469081411769460072514533, 2796695202810912964316564315201939748596257179103, 10992346023695223656990113305938008665991355909256, 43213925872745261764191623041326797702692338171465, 169919458666295781125747001814691224109039528466759, 668261778639994529605542221490462147627480905371952, 2628647547920156303300407597321716893528270042678697, 10341856486590710638008344904097519877779634504984339, 40695210989716085779821412429247323519330798417777320, 160164052674362935854157238299175381813856594604069421, 630466613215316542770297364774862366706965926961535743, 2482177531871398498068830277398736945842068294216635908, 9774076746841723688469820974528620529058379386759469931, 38493669196860579721407641503059896463048677039202716635, 151625443406700808240607986058830864379273846923241820774, 597341461619823330850133772271253155678465525760721540467, 2353638039731486787157223469532728454051610092053465993567
Generating function in Maple syntax:
1/2*((-x^2+5*x-2)*(1-4*x)^(1/2)+3*x^2-7*x+2)/(x^2-3*x+1)/x
Generating function in latex syntax:
\frac{\left(-x^{2}+5 x -2\right) \sqrt{1-4 x}+3 x^{2}-7 x +2}{2 \left(x^{2}-3 x +1\right) x}
Generating function in sympy syntax:
(3*x**2/2 - 7*x/2 + sqrt(1 - 4*x)*(-x**2 + 5*x - 2)/2 + 1)/(x*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)^2*x*F(x)^2-(x-2)*(3*x-1)*(x^2-3*x+1)*F(x)+x^4-8*x^3+21*x^2-12*x+2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right)^{2} x F \! \left(x \right)^{2}-\left(x -2\right) \left(3 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+x^{4}-8 x^{3}+21 x^{2}-12 x +2 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(n+5) = (1+2*n)/(n+6)*a(n)-1/2*(58+33*n)/(n+6)*a(1+n)+(107+40*n)/(n+6)*a(n+2)-(116+31*n)/(n+6)*a(n+3)+1/2*(92+19*n)/(n+6)*a(n+4), n >= 5
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(n +5\right) = \frac{\left(1+2 n \right) a \! \left(n \right)}{n +6}-\frac{\left(58+33 n \right) a \! \left(1+n \right)}{2 \left(n +6\right)}+\frac{\left(107+40 n \right) a \! \left(n +2\right)}{n +6}-\frac{\left(116+31 n \right) a \! \left(n +3\right)}{n +6}+\frac{\left(92+19 n \right) a \! \left(n +4\right)}{2 n +12}, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/333/
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[12,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[12,x]*F[18,x]*F[7,x]
F[7,x] = F[4,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[13,x]*F[18,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[10,x]*F[12,x]
F[12,x] = x
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[15,x]
F[15,x] = F[1,x]+F[14,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[10,x]*F[12,x]*F[15,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]^2*F[12,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_{12}\! \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_{12}\! \left(x \right) F_{18}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{4}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{18}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{15}\! \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_{18} \left(x \right)^{2} F_{12}\! \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_12(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_12(x)*F_18(x)*F_7(x))
Eq(F_7(x), F_4(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_13(x)*F_18(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_10(x)*F_12(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_15(x))
Eq(F_15(x), F_1(x) + F_14(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_10(x)*F_12(x)*F_15(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_18(x)**2)
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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [2, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [1, 0], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], 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