0231_0312_1032_1302_3021
Counting sequence:
1, 1, 2, 6, 19, 59, 186, 602, 1996, 6752, 23221, 80962, 285541, 1016912, 3651938, 13210050, 48087540, 176028523, 647568030, 2392830397, 8877060016, 33051518556, 123462678807, 462570272241, 1737836125894, 6545358461879, 24709677311631, 93483437641527, 354380040538576, 1345893782193658, 5120416354688380, 19512098047071169, 74466802049087714, 284604225622762099, 1089189159490730338, 4173641753464572830, 16012025751972546117, 61499062878133483511, 236459030883432805042, 910090505693450882803, 3506165517941469759985, 13520068780484203392475, 52180113063371832006888, 201554761162920431098190, 779159824284935779977102, 3014325158850810466558886, 11669958831239827939672701, 45211692788002389334482862, 175275741683318159883124011, 679940042611234268595710852, 2639283551726460225454470646, 10250797120674306886504818077, 39835980418912544621933203451, 154892488740147907622824211652, 602578164242349552775081639347, 2345396624948084606789678209141, 9133373345619381578352341021103, 35583665611113100156956513029770, 138697186967777941038371643699208, 540848228788054182509341130379888, 2109927791288868281402580031310100, 8234517186686600432664984522652161, 32150023420580814206475786469695082, 125571639858636160706549869798983086, 490640876720033511432235042445496475, 1917753534896861228941134815266826783, 7498493132117247864508042231935273838, 29329374369483460871033208640676914695, 114755853897222956370973858429663304623, 449144401499281099159993307730656283581, 1758458995913572036702388736130248700002, 6886678427994527134890730912211494536782, 26978327006688385784882653083790106755076, 105716889333365701023517940650884301500322, 414375938193786407345080096096405096163863, 1624659109371605587612147715176089201969210, 6371540405124467079664213336777184184994705, 24994135939885220378422801707482862170717718, 98070964498346514716414968822857590854833450, 384900650842195742892596243475390802221870787, 1510984728827718533431755733867911164020219971, 5932970432615358175287832599196182407405337266, 23301426791256053776877176057157550762456192717, 91535318651643251956795366592797956420757845261, 359656943731376820862357945269950703270732556275, 1413447122846498193469886724488255823414404591788, 5555970373916785045722849388175694693063230114228, 21843763292572397753042748411029188379609073818160, 85897436747556782887702002405053571402936234724628, 337844017499173650926009281350982150701003946832391, 1329026667082400849310068125817765781672151235540352, 5229146390044345826855490461696395771860905559097987, 20578125656978814202441933761439762860083973341762306, 80994786084949333655639583206617916352941526324957682, 318847429832318987441799230761043835066907577419504243, 1255399110093027939298567180887488418345872229156429475, 4943701356949979701433832205986406896663392292984769910, 19471198247330039069929535277533155406355964663454817233, 76701125870698050974490298137155055764052996817033216165, 302188561593866713078799745405335077853142173407920825447, 1190748878077393546704769672515273551596058689574731512876
Generating function in Maple syntax:
1/2*((-2*x^3+4*x^2-3*x+1)*(1-4*x)^(1/2)+2*x^6-2*x^5+4*x^4-2*x^3-2*x^2+3*x-1)/x/(x^4-2*x^3+4*x^2-3*x+1)/(x-1)
Generating function in latex syntax:
\frac{\left(-2 x^{3}+4 x^{2}-3 x +1\right) \sqrt{1-4 x}+2 x^{6}-2 x^{5}+4 x^{4}-2 x^{3}-2 x^{2}+3 x -1}{2 x \left(x^{4}-2 x^{3}+4 x^{2}-3 x +1\right) \left(x -1\right)}
Generating function in sympy syntax:
(x**6 - x**5 + 2*x**4 - x**3 - x**2 + 3*x/2 + sqrt(1 - 4*x)*(-2*x**3 + 4*x**2 - 3*x + 1)/2 - 1/2)/(x*(x - 1)*(x**4 - 2*x**3 + 4*x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
x*(x^4-2*x^3+4*x^2-3*x+1)*(x-1)^2*F(x)^2-(x-1)*(2*x^6-2*x^5+4*x^4-2*x^3-2*x^2+3*x-1)*F(x)+x^7+x^5-x^4-3*x^3+6*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x^{4}-2 x^{3}+4 x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)^{2}-\left(x -1\right) \left(2 x^{6}-2 x^{5}+4 x^{4}-2 x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)+x^{7}+x^{5}-x^{4}-3 x^{3}+6 x^{2}-4 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) = 186
a(7) = 602
a(8) = 1996
a(9) = 6752
a(n+7) = 4*(5+2*n)/(n+8)*a(n)-2*(42+13*n)/(n+8)*a(n+1)+2*(103+29*n)/(n+8)*a(n+2)-(320+77*n)/(n+8)*a(n+3)+64*(n+5)/(n+8)*a(n+4)-2*(95+16*n)/(n+8)*a(n+5)+(62+9*n)/(n+8)*a(n+6)+3*(n+2)/(n+8), n >= 10
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) = 186
a \! \left(7\right) = 602
a \! \left(8\right) = 1996
a \! \left(9\right) = 6752
a \! \left(n +7\right) = \frac{4 \left(5+2 n \right) a \! \left(n \right)}{n +8}-\frac{2 \left(42+13 n \right) a \! \left(n +1\right)}{n +8}+\frac{2 \left(103+29 n \right) a \! \left(n +2\right)}{n +8}-\frac{\left(320+77 n \right) a \! \left(n +3\right)}{n +8}+\frac{64 \left(n +5\right) a \! \left(n +4\right)}{n +8}-\frac{2 \left(95+16 n \right) a \! \left(n +5\right)}{n +8}+\frac{\left(62+9 n \right) a \! \left(n +6\right)}{n +8}+\frac{3 n +6}{n +8}, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21976/
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[7,x]
F[7,x] = F[13,x]+F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[9,x]^2*F[12,x]
F[12,x] = x
F[13,x] = F[14,x]
F[14,x] = F[15,x]*F[18,x]
F[15,x] = F[16,x]+F[2,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[12,x]*F[16,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[1,x]+F[19,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_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{13}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{9} \left(x \right)^{2} 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_{15}\! \left(x \right) F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{2}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \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_7(x))
Eq(F_7(x), F_13(x) + F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x)**2)
Eq(F_12(x), x)
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_15(x)*F_18(x))
Eq(F_15(x), F_16(x) + F_2(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_12(x)*F_16(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_19(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "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, 0, 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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "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, 0, 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": [[{"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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "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, 0, 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_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], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "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, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 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], [2, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "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, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "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, 0, 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": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], 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