0123_0213_1230_1302
Counting sequence:
1, 1, 2, 6, 20, 64, 194, 564, 1592, 4402, 11996, 32352, 86594, 230500, 611048, 1614930, 4258316, 11209168, 29467490, 77389908, 203095448, 532682866, 1396526012, 3660040896, 9589888130, 25122206404, 65801896904, 172333815954, 451300214252, 1181768153392, 3094406899106, 8102257850292, 21213977264504, 55542895168690, 145421150692508, 380733441810720, 996804944543426, 2609732931427108, 6832496928953000, 17887964013862098, 46831807429493708, 122608282908339856, 320994690562967522, 840379087315446036, 2200149168453137240, 5760081612183498994, 15080122056376426364, 39480337333503913344, 103360995497251580162, 270602860264483360132, 708448007508663566216, 1854742006686437470482, 4855779701400509109164, 12712600475214810384880, 33282028479643363101218, 87133498474514161030260, 228118493965496884212536, 597222037465172020053298, 1563547726516410232839260, 4093421358256840792248288, 10716716780599676371473218, 28056729848233316777306596, 73453474493482530870717032, 192303697090978789655385426, 503457623696982865736521100, 1318069187835027862836341584, 3450749967478216833336831074, 9034180769939854858302806484, 23651792453021812183828898072, 61921196810486510577698507122, 162111798421159577318295862076, 424414199338435936915247556672, 1111130801365035664503563763074, 2908978208298445918747677642820, 7615803830613851816043936985928, 19938433297710208977993068956050, 52199496090850974015153141164396, 136660055031511110861902097101488, 357780669117019154159424635268770, 936681952546219942794114778962228, 2452265188974987856578405642132728, 6420113615285437991652074028465586, 16808075658694714847799760205323292, 44004113364425484010591094111622816, 115204264441835292101661297177782210, 301608679975587502129768347518197924, 789621775513941433958394845569759784, 2067256646624265239086918389576977874, 5412148164474911161985364723932966732, 14169187847032582004235184583765508112, 37095415377087062365452206630450729186, 97117058285157060121585470513761022612, 254255759480241028058232275323181024984, 665650220159279844170967496280479425010, 1742694901005026144690382495167651995388, 4562434482870653870371604552521266051840, 11944608547636646027367280288993725141506, 31271391160098705333615934567655067335428, 81869564932778312217251919920361792790280, 214337303638473915805682618206210942886418, 561142345983118804174881520723832299570988
Generating function in Maple syntax:
(x^4-3*x^3+8*x^2-5*x+1)/(2*x-1)/(x-1)/(x^2-3*x+1)
Generating function in latex syntax:
\frac{x^{4}-3 x^{3}+8 x^{2}-5 x +1}{\left(2 x -1\right) \left(x -1\right) \left(x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(x**4 - 3*x**3 + 8*x**2 - 5*x + 1)/((x - 1)*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)*(x^2-3*x+1)*F(x)-x^4+3*x^3-8*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)-x^{4}+3 x^{3}-8 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,2-2/5*(3/2+1/2*5^(1/2))^(-n)*5^(1/2)+2/5*(3/2-1/2*5^(1/2))^(-n)*5^(1/2)-3*2^(n-1))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 2-\frac{2 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}+\frac{2 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}-3 \,2^{n -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(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)+2, 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) = 20
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+2, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14924/
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[16,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[13,x]
F[16,x] = F[17,x]+F[28,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[20,x]+F[9,x]
F[20,x] = F[17,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[27,x]
F[22,x] = 0
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[10,x]
F[26,x] = F[21,x]
F[27,x] = F[12,x]*F[20,x]
F[28,x] = F[22,x]+F[29,x]+F[53,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[31,x]+F[34,x]
F[31,x] = F[32,x]+F[6,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[25,x]
F[34,x] = F[28,x]+F[35,x]
F[35,x] = 2*F[22,x]+F[36,x]+F[40,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[32,x]
F[39,x] = F[35,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[22,x]+F[27,x]+F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[45,x]+F[47,x]
F[45,x] = F[10,x]+F[46,x]
F[46,x] = F[33,x]
F[47,x] = F[42,x]+F[48,x]
F[48,x] = 2*F[22,x]+F[40,x]+F[49,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[46,x]
F[52,x] = F[48,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[17,x]+F[42,x]
F[56,x] = F[57,x]+F[63,x]
F[57,x] = F[22,x]+F[23,x]+F[58,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[20,x]+F[60,x]
F[60,x] = F[57,x]+F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[12,x]*F[60,x]
F[63,x] = F[62,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_{16}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \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_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{28}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{9}\! \left(x \right)
F_{20}\! \left(x \right) = F_{17}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right)
F_{26}\! \left(x \right) = F_{21}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{28}\! \left(x \right) = F_{22}\! \left(x \right)+F_{29}\! \left(x \right)+F_{53}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{6}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{34}\! \left(x \right) = F_{28}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{32}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{10}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{33}\! \left(x \right)
F_{47}\! \left(x \right) = F_{42}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{40}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{46}\! \left(x \right)
F_{52}\! \left(x \right) = F_{48}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{17}\! \left(x \right)+F_{42}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{63}\! \left(x \right)
F_{57}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{20}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{57}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{62}\! \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_16(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(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))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_13(x))
Eq(F_16(x), F_17(x) + F_28(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_20(x) + F_9(x))
Eq(F_20(x), F_17(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_27(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_10(x))
Eq(F_26(x), F_21(x))
Eq(F_27(x), F_12(x)*F_20(x))
Eq(F_28(x), F_22(x) + F_29(x) + F_53(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_34(x))
Eq(F_31(x), F_32(x) + F_6(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_25(x))
Eq(F_34(x), F_28(x) + F_35(x))
Eq(F_35(x), 2*F_22(x) + F_36(x) + F_40(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_32(x))
Eq(F_39(x), F_35(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_22(x) + F_27(x) + F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_47(x))
Eq(F_45(x), F_10(x) + F_46(x))
Eq(F_46(x), F_33(x))
Eq(F_47(x), F_42(x) + F_48(x))
Eq(F_48(x), 2*F_22(x) + F_40(x) + F_49(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_46(x))
Eq(F_52(x), F_48(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_17(x) + F_42(x))
Eq(F_56(x), F_57(x) + F_63(x))
Eq(F_57(x), F_22(x) + F_23(x) + F_58(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_20(x) + F_60(x))
Eq(F_60(x), F_57(x) + F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_12(x)*F_60(x))
Eq(F_63(x), F_62(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": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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, 2, 3], "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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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