0132_0231_1032_1320_2031
Counting sequence:
1, 1, 2, 6, 19, 60, 192, 626, 2079, 7023, 24089, 83742, 294533, 1046424, 3750375, 13543656, 49234931, 180027111, 661664398, 2443023955, 9057337932, 33703882867, 125838810817, 471274556476, 1769882988034, 6663876251625, 25149747113522, 95123367831416, 360511249771664, 1368884675656119, 5206862071210883, 19837943826960294, 75697854483897616, 289265037330950627, 1106869891188869918, 4240836117794310759, 16267828867984502179, 62474436308618200604, 240183672526109619096, 924333685898762233539, 3560703908380565693094, 13729160890457559760222, 52982682912335360087208, 204638731966384239828227, 791022822362844741646303, 3060003674518001776569223, 11846010901718366476625879, 45890836729387951900008546, 177897878912942879943526129, 690072261322184738058536818, 2678466137789295861145137239, 10402434394366447882066619987, 40423238613831214827770839408, 157168368264532319284162321244, 611403965816870202393109940259, 2379644384000209723003618117142, 9266349070897208336767216489454, 36100277652949886999427312870468, 140705349983908977875396431523245, 548658508127617000443064662714393, 2140319752063299861063560974087793, 8352839671422587091211622764081821, 32610899899204200575469719551784792, 127367626936599566732399037580254626, 497642783890379632237257248075073115, 1945063264211645421914769347948515104, 7605054578086742349882319586479348524, 29745341046012754298615149555080373159, 116380231721502333452382731133184940122, 455490112055997686090575397291765200320, 1783257931032996543511099664043103661658, 6983626790798526609738750785951024339663, 27357465543311925947515769227945195168243, 107200093445012843951357123004449552465909, 420180175867763308016732596876823937458735, 1647380067972119020517145819665204909544626, 6460509956935936364750530132549185078144561, 25342622235557290225048208207408731164544456, 99436352142761818531363766561981929397706793, 390251820962442021800596410330597035204972365, 1531962569030024572640229454190470724855687882, 6015230459220651094311944472019163469190810532, 23624075501442720848071196862757800167372518077, 92801165344409582627153269703026696715274049536, 364624463076571031326153636102424003430461534250, 1432945685344039077762985984978373762313293916744, 5632524321398104696570781073639399159640934556019, 22144393103544851144487857719865989769586229392749, 87078283500280218906429604514848972622244920228337, 342483288557004451449110647403486020928100977758645, 1347257160415190033901611263992531755517659098635698, 5300799887331983903911007974930647239137198641754843, 20859811207111511388094852451943241938427422852247298, 82102373292584624325210334723276369908740601270170289, 323203306018296920163696862863411546470820503703612495, 1272532980181703060738200365420718854711964188222624796, 5011110054676332567741704091333195950787562844667514416, 19736448108532260871066453823337017766624688347250014021, 77745056039335840454391072977183736460914143655801821808, 306297817002202979596111388212901271113816973096111474790, 1206927019500670742800593101094568096795329294323048033452
Generating function in Maple syntax:
1/4*((x-1)^4*(-4*x+1)^(1/2)+2*x^6-2*x^5+5*x^4-2*x^3-4*x^2+4*x-1)/(x-1/2)/x/(x^4-2*x^3+4*x^2-3*x+1)
Generating function in latex syntax:
\frac{\left(x -1\right)^{4} \sqrt{-4 x +1}+2 x^{6}-2 x^{5}+5 x^{4}-2 x^{3}-4 x^{2}+4 x -1}{4 \left(x -\frac{1}{2}\right) x \left(x^{4}-2 x^{3}+4 x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(x**6/2 - x**5/2 + 5*x**4/4 - x**3/2 - x**2 + x + sqrt(1 - 4*x)*(x - 1)**4/4 - 1/4)/(x*(x - 1/2)*(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)*(2*x-1)^2*F(x)^2-(2*x-1)*(2*x^6-2*x^5+5*x^4-2*x^3-4*x^2+4*x-1)*F(x)+x^7+2*x^5+x^4-11*x^3+13*x^2-6*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(2 x -1\right)^{2} F \! \left(x \right)^{2}-\left(2 x -1\right) \left(2 x^{6}-2 x^{5}+5 x^{4}-2 x^{3}-4 x^{2}+4 x -1\right) F \! \left(x \right)+x^{7}+2 x^{5}+x^{4}-11 x^{3}+13 x^{2}-6 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 60
a(6) = 192
a(7) = 626
a(n+7) = 4*(3+2*n)/(n+8)*a(n)-2*(44+15*n)/(n+8)*a(1+n)+(221+67*n)/(n+8)*a(n+2)-5*(78+19*n)/(n+8)*a(n+3)+80*(n+5)/(n+8)*a(n+4)-3*(77+13*n)/(n+8)*a(n+5)+(69+10*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) = 19
a \! \left(5\right) = 60
a \! \left(6\right) = 192
a \! \left(7\right) = 626
a \! \left(n +7\right) = \frac{4 \left(3+2 n \right) a \! \left(n \right)}{n +8}-\frac{2 \left(44+15 n \right) a \! \left(1+n \right)}{n +8}+\frac{\left(221+67 n \right) a \! \left(n +2\right)}{n +8}-\frac{5 \left(78+19 n \right) a \! \left(n +3\right)}{n +8}+\frac{80 \left(n +5\right) a \! \left(n +4\right)}{n +8}-\frac{3 \left(77+13 n \right) a \! \left(n +5\right)}{n +8}+\frac{\left(69+10 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21701/
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[12,x]*F[15,x]
F[15,x] = F[16,x]+F[22,x]
F[16,x] = F[17,x]*F[19,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[12,x]*F[17,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[12,x]*F[17,x]*F[19,x]
F[22,x] = F[2,x]*F[23,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[12,x]*F[23,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_{12}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{22}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{19}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \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_12(x)*F_15(x))
Eq(F_15(x), F_16(x) + F_22(x))
Eq(F_16(x), F_17(x)*F_19(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_12(x)*F_17(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_12(x)*F_17(x)*F_19(x))
Eq(F_22(x), F_2(x)*F_23(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_12(x)*F_23(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]]}, {"patt": [2, 0, 3, 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, 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": [2, 0, 3, 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, 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": [2, 0, 3, 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], [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": [1, 0, 2], "pos": [[0, 1], [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]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [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": [1, 0, 2], "pos": [[0, 2], [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]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], 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