0213_1032_1230_1320
Counting sequence:
1, 1, 2, 6, 20, 62, 181, 507, 1381, 3690, 9728, 25404, 65897, 170129, 437797, 1124136, 2882500, 7385682, 18918401, 48462207, 124182761, 318378470, 816793228, 2097057264, 5388496237, 13858052829, 35671682657, 91903977276, 236990873720, 611658154862, 1579995796861, 4084708277667, 10568429563981, 27364620987810, 70905936026168, 183854474821284, 477033050332817, 1238482055312489, 3217233370999357, 8362035818068176, 21745209600280780, 56574841259139402, 147257665210606361, 383455142643308487, 998898166148697041, 2603073302641470110, 6785772342301071748, 17694963478988360664, 46155977418317022037, 120427345605344189109, 314291549234829213497, 820432412312147649636, 2142140905383526556480, 5594244864252843372902, 14612236055560902114901, 38173998658566397481067, 99744465119297910001141, 260660326691184410580570, 681273048611931705936368, 1780832753401425770491404, 4655609487662862549885497, 12172437943783431698375489, 31828972583747330862513877, 83235457512968310811931256, 217686621588624366674100820, 569363824190122803210813762, 1489292898419196303812117681, 3895830366738687579437678607, 10191717208294959508787258681, 26663674702249994308725487190, 69761532664033514699311461628, 182529728006856918682921082304, 477607486039619636354930967997, 1249761203162155481158807481229, 3270392902164200847425372196497, 8558319529683237959991066647436, 22397086510044307525577344421000, 58614883656591781811629999257182, 153403972562117461412032718813101, 401491793824562415864622287757107, 1050817336471333575044512405087261, 2750346830239315177393725447370770, 7198742310577224436167562973452328, 18842305027745649865132771537838964, 49319541779381300921983665750461537, 129095483245863455791696659887720729, 337916602827398567260313692148504557, 884532877893251489019932798757689376, 2315388831001533963718859229868387420, 6060925767998925927107663090751471162, 15865679578784173239668702500370754761, 41531987332548228777404930591100065367, 108720322252497113718436280222124347681, 284604933455327751080816070413865328910, 745036426005725622486977027164777780468, 1950364194372379930871458532747525711784, 5105717804616049701798053048632927025317, 13365972364278239891805357766079980665189, 34990227225293584618753998385322417764457, 91599948330485489329811269518852452561076, 239798123740865099462125089974092000665680
Generating function in Maple syntax:
-(x^7+2*x^6-x^5-x^4-14*x^3+17*x^2-7*x+1)/(x-1)/(2*x-1)/(x^2-3*x+1)/(x^2+2*x-1)
Generating function in latex syntax:
-\frac{x^{7}+2 x^{6}-x^{5}-x^{4}-14 x^{3}+17 x^{2}-7 x +1}{\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+2 x -1\right)}
Generating function in sympy syntax:
(-x**7 - 2*x**6 + x**5 + x**4 + 14*x**3 - 17*x**2 + 7*x - 1)/((x - 1)*(2*x - 1)*(x**2 - 3*x + 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x-1)*(x^2-3*x+1)*(x^2+2*x-1)*F(x)+x^7+2*x^6-x^5-x^4-14*x^3+17*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+2 x -1\right) F \! \left(x \right)+x^{7}+2 x^{6}-x^{5}-x^{4}-14 x^{3}+17 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,(3/2-1/2*5^(1/2))^(1-n)+(3/2+1/2*5^(1/2))^(1-n)+1/4*(2-2^(1/2))*(-1-2^(1/2))^(-n)+1/4*(2+2^(1/2))*(2^(1/2)-1)^(-n)-7/4*2^n+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{1-n}+\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{1-n}+\frac{\left(2-\sqrt{2}\right) \left(-1-\sqrt{2}\right)^{-n}}{4}+\\\frac{\left(2+\sqrt{2}\right) \left(\sqrt{2}-1\right)^{-n}}{4}-\frac{7 \,2^{n}}{4}+1 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 62
a(6) = 181
a(7) = 507
a(n+5) = -2*a(n)+3*a(n+1)+11*a(n+2)-16*a(n+3)+7*a(n+4)-2, 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) = 20
a \! \left(5\right) = 62
a \! \left(6\right) = 181
a \! \left(7\right) = 507
a \! \left(n +5\right) = -2 a \! \left(n \right)+3 a \! \left(n +1\right)+11 a \! \left(n +2\right)-16 a \! \left(n +3\right)+7 a \! \left(n +4\right)-2, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21374/
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[17,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[17,x]*F[7,x]
F[7,x] = F[51,x]+F[8,x]
F[8,x] = F[33,x]+F[9,x]
F[9,x] = F[10,x]+F[21,x]
F[10,x] = F[11,x]*F[18,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[11,x]*F[15,x]*F[17,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[15,x]*F[17,x]
F[17,x] = x
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[11,x]*F[17,x]*F[18,x]
F[21,x] = F[15,x]*F[22,x]*F[24,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[17,x]*F[22,x]
F[24,x] = F[25,x]
F[25,x] = F[17,x]*F[26,x]
F[26,x] = F[1,x]+F[27,x]+F[31,x]
F[27,x] = F[28,x]
F[28,x] = F[17,x]*F[22,x]*F[29,x]
F[29,x] = F[1,x]+F[30,x]
F[30,x] = F[24,x]
F[31,x] = F[32,x]
F[32,x] = F[17,x]*F[22,x]*F[26,x]
F[33,x] = F[15,x]*F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[15,x]*F[17,x]*F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[17,x]*F[38,x]
F[38,x] = F[39,x]+F[49,x]
F[39,x] = F[33,x]+F[40,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[42,x]
F[42,x] = F[15,x]*F[17,x]*F[22,x]*F[43,x]
F[43,x] = F[19,x]
F[44,x] = F[45,x]+F[47,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[13,x]*F[18,x]
F[47,x] = F[22,x]*F[24,x]*F[48,x]
F[48,x] = F[16,x]
F[49,x] = F[15,x]*F[43,x]*F[50,x]
F[50,x] = F[23,x]
F[51,x] = F[15,x]*F[52,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[22,x]*F[24,x]
F[54,x] = F[43,x]*F[50,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_{17}\! \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_{17}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{51}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{33}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{18}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right) F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{15}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = x
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_{11}\! \left(x \right) F_{17}\! \left(x \right) F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right) F_{22}\! \left(x \right) F_{24}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{17}\! \left(x \right) F_{22}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{17}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{17}\! \left(x \right) F_{22}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{24}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{17}\! \left(x \right) F_{22}\! \left(x \right) F_{26}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right) F_{17}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{17}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{49}\! \left(x \right)
F_{39}\! \left(x \right) = F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right) F_{17}\! \left(x \right) F_{22}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{19}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \left(x \right)
F_{47}\! \left(x \right) = F_{22}\! \left(x \right) F_{24}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{16}\! \left(x \right)
F_{49}\! \left(x \right) = F_{15}\! \left(x \right) F_{43}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{23}\! \left(x \right)
F_{51}\! \left(x \right) = F_{15}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{22}\! \left(x \right) F_{24}\! \left(x \right)
F_{54}\! \left(x \right) = F_{43}\! \left(x \right) F_{50}\! \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_17(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_17(x)*F_7(x))
Eq(F_7(x), F_51(x) + F_8(x))
Eq(F_8(x), F_33(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_21(x))
Eq(F_10(x), F_11(x)*F_18(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_11(x)*F_15(x)*F_17(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_15(x)*F_17(x))
Eq(F_17(x), x)
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_11(x)*F_17(x)*F_18(x))
Eq(F_21(x), F_15(x)*F_22(x)*F_24(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_17(x)*F_22(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_17(x)*F_26(x))
Eq(F_26(x), F_1(x) + F_27(x) + F_31(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_17(x)*F_22(x)*F_29(x))
Eq(F_29(x), F_1(x) + F_30(x))
Eq(F_30(x), F_24(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_17(x)*F_22(x)*F_26(x))
Eq(F_33(x), F_15(x)*F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_15(x)*F_17(x)*F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_17(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_49(x))
Eq(F_39(x), F_33(x) + F_40(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_15(x)*F_17(x)*F_22(x)*F_43(x))
Eq(F_43(x), F_19(x))
Eq(F_44(x), F_45(x) + F_47(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_13(x)*F_18(x))
Eq(F_47(x), F_22(x)*F_24(x)*F_48(x))
Eq(F_48(x), F_16(x))
Eq(F_49(x), F_15(x)*F_43(x)*F_50(x))
Eq(F_50(x), F_23(x))
Eq(F_51(x), F_15(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_22(x)*F_24(x))
Eq(F_54(x), F_43(x)*F_50(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, 1, 3], "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, 2, 3, 0], "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, 2, 1, 3], "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, 2, 3, 0], "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": 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