0132_0213_1320
Counting sequence:
1, 1, 2, 6, 21, 77, 283, 1032, 3740, 13522, 48930, 177564, 646908, 2367121, 8699706, 32108614, 118975273, 442467434, 1651076429, 6180073782, 23197995681, 87304824502, 329357941951, 1245262325246, 4717865597299, 17908489016748, 68099538024617, 259388123347450, 989533568712759, 3780451457411265, 14462769670397403, 55401214350006496, 212478710610645660, 815850972601255742, 3136020019633562962, 12066874559331807444, 46476780661930934528, 179176697693352782602, 691371988756382226584, 2669984176792016857320, 10319422638342519152202, 39915006546173119285256, 154502827331235651345554, 598469959635079355449320, 2319749682104845881279762, 8997475596859425595212286, 34919634544073009319678754, 135605563650780048384802292, 526907160101107473025226780, 2048471882228570585663183856, 7968127973865457302502292088, 31010198308735273989444845116, 120744332487274130449574537428, 470365410310638802763270032352, 1833172132458838098935518353140, 7147654747742926180836531227220, 27881098662898350629384747393472, 108801795369111642344901023351476, 424752815951429065784714728592752, 1658841692447136617222648671451348, 6480921994982039731244141451289232, 25329509229571272032171639814296841, 99030749960889500681212529504052618, 387312998513100703785128331625498630, 1515299195931413926899008610756531961, 5930274892507428554409585927101622434, 23215997430431839944747295659307125341, 90914280632685478112079537512320713366, 356127420277128806638309132227258432777, 1395416102778054680599833383277485915410, 5469198204958881988830936271554750941887, 21441835171777819159389983318958296867702, 84084408588031976565483712432483480842983, 329823213725831954653923096945400875920196, 1294065619231992591079859161384235650789533, 5078528973001311867363217651501693863856018, 19935330986676979353752468788930622666629579, 78272692338676467236697367119929813100217082, 307394377262889275619457053288957683121356449, 1207474590950231123025453432307271701567036194, 4744104115158014385339706014904406411845354325, 18643280295573858225721873666412799563353446988, 73279113683639570479604687441612989729772176575, 288088403511919678484978057617283744429364936870, 1132809900477103005091346484449223614143864075401, 4455249731855899069844636898135062280964949071864, 17525441169252438547781888071476881895497546419071, 68951854249290159952822093433368475808309537924026, 271332163033502175130295303351691837499562675330857, 1067906551446870188073434725691993572726635900720720, 4203782472251754556438468324097902504589080667134127, 16550863054449879728800621554830331001069059463366822, 65173785468854445442092626916989334732747817945481093, 256682140598575828026768984757624576775761180045470966, 1011084193405906400618010860867293374561415054394743143, 3983332455056167405074560352834284062902731166491589878, 15695385911128338316244858922514671789460136834852529179, 61853223316950407467535753733202165243615430282913669852, 243790227852544014238589409921475763470358845518383899569, 961020449715869451342537986817315312132720804620658094834, 3788873811005963527061920668057348128845455854560429491895
Generating function in Maple syntax:
1/4*((-8*x^5+26*x^4-37*x^3+25*x^2-8*x+1)*(1-4*x)^(1/2)-4*x^6+28*x^5-66*x^4+73*x^3-39*x^2+10*x-1)/(x-1)^2/x^3/(x-1/2)
Generating function in latex syntax:
\frac{\left(-8 x^{5}+26 x^{4}-37 x^{3}+25 x^{2}-8 x +1\right) \sqrt{1-4 x}-4 x^{6}+28 x^{5}-66 x^{4}+73 x^{3}-39 x^{2}+10 x -1}{4 \left(x -1\right)^{2} x^{3} \left(x -\frac{1}{2}\right)}
Generating function in sympy syntax:
(-x**6 + 7*x**5 - 33*x**4/2 + 73*x**3/4 - 39*x**2/4 + 5*x/2 + sqrt(1 - 4*x)*(-8*x**5 + 26*x**4 - 37*x**3 + 25*x**2 - 8*x + 1)/4 - 1/4)/(x**3*(x - 1)**2*(x - 1/2))
Implicit equation for the generating function in Maple syntax:
x^3*(2*x-1)^2*(x-1)^4*F(x)^2+(2*x-1)*(4*x^6-28*x^5+66*x^4-73*x^3+39*x^2-10*x+1)*(x-1)^2*F(x)+4*x^9+8*x^8-104*x^7+302*x^6-452*x^5+403*x^4-220*x^3+72*x^2-13*x+1 = 0
Implicit equation for the generating function in latex syntax:
x^{3} \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)^{2}+\left(2 x -1\right) \left(4 x^{6}-28 x^{5}+66 x^{4}-73 x^{3}+39 x^{2}-10 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+4 x^{9}+8 x^{8}-104 x^{7}+302 x^{6}-452 x^{5}+403 x^{4}-220 x^{3}+72 x^{2}-13 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 77
a(6) = 283
a(7) = 1032
a(8) = 3740
a(9) = 13522
a(n+8) = -32*(2*n+1)/(11+n)*a(n)+2*(454*n+1935)/(11+n)*a(3+n)+8*(40*n+67)/(11+n)*a(n+1)-4*(179*n+532)/(11+n)*a(n+2)-3*(233*n+1294)/(11+n)*a(n+4)+(2281+332*n)/(11+n)*a(n+5)-(95*n+781)/(11+n)*a(n+6)+3*(5*n+48)/(11+n)*a(n+7)+4/(11+n), 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) = 21
a \! \left(5\right) = 77
a \! \left(6\right) = 283
a \! \left(7\right) = 1032
a \! \left(8\right) = 3740
a \! \left(9\right) = 13522
a \! \left(n +8\right) = -\frac{32 \left(2 n +1\right) a \! \left(n \right)}{11+n}+\frac{2 \left(454 n +1935\right) a \! \left(3+n \right)}{11+n}+\frac{8 \left(40 n +67\right) a \! \left(n +1\right)}{11+n}-\frac{4 \left(179 n +532\right) a \! \left(n +2\right)}{11+n}-\frac{3 \left(233 n +1294\right) a \! \left(n +4\right)}{11+n}+\frac{\left(2281+332 n \right) a \! \left(n +5\right)}{11+n}-\frac{\left(95 n +781\right) a \! \left(n +6\right)}{11+n}+\frac{3 \left(5 n +48\right) a \! \left(n +7\right)}{11+n}+\frac{4}{11+n}, \quad n \geq 10
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/258/
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[7,x]
F[3,x] = F[4,x]+F[8,x]
F[4,x] = F[1,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[4,x]^2*F[7,x]
F[7,x] = x
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[7,x]
F[10,x] = F[11,x]+F[42,x]
F[11,x] = F[12,x]+F[14,x]+F[4,x]
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[4,x]*F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[7,x]
F[16,x] = F[17,x]+F[26,x]
F[17,x] = F[18,x]+F[19,x]+F[24,x]
F[18,x] = F[4,x]^2
F[19,x] = F[20,x]
F[20,x] = F[17,x]*F[21,x]*F[7,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[21,x]*F[7,x]
F[24,x] = F[25,x]
F[25,x] = F[17,x]*F[4,x]*F[7,x]
F[26,x] = F[27,x]
F[27,x] = F[21,x]*F[28,x]*F[32,x]*F[35,x]*F[7,x]
F[28,x] = F[29,x]+F[30,x]+F[4,x]
F[29,x] = F[28,x]*F[7,x]
F[30,x] = F[31,x]
F[31,x] = F[28,x]*F[4,x]*F[7,x]
F[32,x] = F[1,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[32,x]*F[7,x]
F[35,x] = F[1,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[7,x]
F[38,x] = F[21,x]+F[39,x]
F[39,x] = F[36,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[39,x]*F[7,x]
F[42,x] = F[43,x]
F[43,x] = F[44,x]*F[7,x]
F[44,x] = F[10,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]^5*F[7,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_{7}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right)+F_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{1}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{4} \left(x \right)^{2} F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = x
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{7}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{42}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{14}\! \left(x \right)+F_{4}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{7}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{26}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4} \left(x \right)^{2}
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{17}\! \left(x \right) F_{21}\! \left(x \right) F_{7}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{21}\! \left(x \right) F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{21}\! \left(x \right) F_{28}\! \left(x \right) F_{32}\! \left(x \right) F_{35}\! \left(x \right) F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{28}\! \left(x \right) F_{7}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{32}\! \left(x \right) = F_{1}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{32}\! \left(x \right) F_{7}\! \left(x \right)
F_{35}\! \left(x \right) = F_{1}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{7}\! \left(x \right)
F_{38}\! \left(x \right) = F_{21}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{39}\! \left(x \right) F_{7}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right) F_{7}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4} \left(x \right)^{5} F_{7}\! \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)*F_7(x))
Eq(F_3(x), F_4(x) + F_8(x))
Eq(F_4(x), F_1(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_4(x)**2*F_7(x))
Eq(F_7(x), x)
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_7(x))
Eq(F_10(x), F_11(x) + F_42(x))
Eq(F_11(x), F_12(x) + F_14(x) + F_4(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_4(x)*F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_7(x))
Eq(F_16(x), F_17(x) + F_26(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_24(x))
Eq(F_18(x), F_4(x)**2)
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_17(x)*F_21(x)*F_7(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_21(x)*F_7(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_17(x)*F_4(x)*F_7(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_21(x)*F_28(x)*F_32(x)*F_35(x)*F_7(x))
Eq(F_28(x), F_29(x) + F_30(x) + F_4(x))
Eq(F_29(x), F_28(x)*F_7(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_28(x)*F_4(x)*F_7(x))
Eq(F_32(x), F_1(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_32(x)*F_7(x))
Eq(F_35(x), F_1(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_7(x))
Eq(F_38(x), F_21(x) + F_39(x))
Eq(F_39(x), F_36(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_39(x)*F_7(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_44(x)*F_7(x))
Eq(F_44(x), F_10(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)**5*F_7(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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": "row_and_col_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": [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], "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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 2], [0, 2], [0, 2]]}, 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