0132_0231_0321_1203_1302
Counting sequence:
1, 1, 2, 6, 19, 57, 165, 473, 1358, 3910, 11273, 32509, 93739, 270269, 779222, 2246618, 6477415, 18675681, 53845789, 155248365, 447612562, 1290557906, 3720940717, 10728228837, 30931666451, 89182288521, 257130685962, 741360093454, 2137492017355, 6162824473049, 17768677119301, 51230712154961, 147708568880054, 425873863624702, 1227881016641857, 3540230851966045, 10207206004037579, 29429452136279973, 84851099576222942, 244642987778461794, 705355520059529087, 2033683508350332577, 5863523421193021853, 16905731284987982277, 48742663710906562042, 140535018898845580554, 405191059192768775685, 1168248282429396099557, 3368297543677874706899, 9711487286891942742865, 28000194193201449231506, 80730258064097912931478, 232761763083710713031107, 671099531365568100118457, 1934916521650129967726501, 5578758098871659096734217, 16084694909310243410401438, 46375448754072735052205622, 133709856436055563558208569, 385512726851629619739058941, 1111511645632901979035710891, 3204714273552495562055979725, 9239843428957673376311001958, 26640348968462818319464815210, 76809547544631780040279605719, 221457556768314000422597205921, 638507204085446604850278760413, 1840946209370202815935569659101, 5307822565054384578194197712098, 15303532628331725326948502908610, 44123198926868065634312471098909, 127216161838065928938714069137189, 366790083820374182916141947273171, 1057530455605218862885303859880025, 3049075517211295345103787514957210, 8791105221018703911368763689167006, 25346545394095828781844604847966459, 73079248543053424634529599318868441, 210702345608857560986633911933284293, 607497741563661521237161802501777409, 1751539618310897978007232116406361606, 5050045168260419973331456837748964334, 14560307934150104496308551364702978801, 41980330882922108367983486309657957085, 121037837181050885536479521030449987531, 348976716508596427570091325398745435957, 1006170892519775839123923127778913325422, 2900995444861159115267180426748483410930, 8364160237262862570218003807734313425775, 24115576120098793677578961809789164716769, 69530113616593345392512785438215288240605, 200469467345927716722731999423073592209973, 577994271080395883728305224524550918186314, 1666475108776930264726441618113838692739322, 4804786876143269716261340283719393493269237, 13853178366463573211261449814218655663142821, 39941532434233185476243321278881703837403155, 115159566345939925285634752733722337460156321, 332028465428095277466476255189897299705811746, 957305635585372220274182094805806675334482726, 2760106964750521370246084540686115953411710067
Generating function in Maple syntax:
(2*x-1)*(x-1)^3/(7*x^4-14*x^3+13*x^2-6*x+1)
Generating function in latex syntax:
\frac{\left(2 x -1\right) \left(x -1\right)^{3}}{7 x^{4}-14 x^{3}+13 x^{2}-6 x +1}
Generating function in sympy syntax:
(x - 1)**3*(2*x - 1)/(7*x**4 - 14*x**3 + 13*x**2 - 6*x + 1)
Implicit equation for the generating function in Maple syntax:
(7*x^4-14*x^3+13*x^2-6*x+1)*F(x)-(2*x-1)*(x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(7 x^{4}-14 x^{3}+13 x^{2}-6 x +1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/56*((-2^(1/2)-1)*(-70+56*2^(1/2))^(1/2)-2^(1/2)+10)*(1/2+1/14*(-35+28*2^(1/2))^(1/2))^(-n)+1/56*((I*2^(1/2)-2*I)*(35+28*2^(1/2))^(1/2)+2^(1/2)+10)*(1/2+1/14*I*(35+28*2^(1/2))^(1/2))^(-n)+1/56*((1+2^(1/2))*(-70+56*2^(1/2))^(1/2)-2^(1/2)+10)*(1/2-1/14*(-35+28*2^(1/2))^(1/2))^(-n)-1/56*((I*2^(1/2)-2*I)*(35+28*2^(1/2))^(1/2)-2^(1/2)-10)*(1/2-1/14*I*(35+28*2^(1/2))^(1/2))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-\sqrt{2}-1\right) \sqrt{-70+56 \sqrt{2}}-\sqrt{2}+10\right) \left(\frac{1}{2}+\frac{\sqrt{-35+28 \sqrt{2}}}{14}\right)^{-n}}{56}+\\\frac{\left(\left(\mathrm{I} \sqrt{2}-2 \,\mathrm{I}\right) \sqrt{35+28 \sqrt{2}}+\sqrt{2}+10\right) \left(\frac{1}{2}+\frac{\mathrm{I} \sqrt{35+28 \sqrt{2}}}{14}\right)^{-n}}{56}+\\\frac{\left(\left(1+\sqrt{2}\right) \sqrt{-70+56 \sqrt{2}}-\sqrt{2}+10\right) \left(\frac{1}{2}-\frac{\sqrt{-35+28 \sqrt{2}}}{14}\right)^{-n}}{56}-\\\frac{\left(\left(\mathrm{I} \sqrt{2}-2 \,\mathrm{I}\right) \sqrt{35+28 \sqrt{2}}-\sqrt{2}-10\right) \left(\frac{1}{2}-\frac{\mathrm{I} \sqrt{35+28 \sqrt{2}}}{14}\right)^{-n}}{56} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+4) = -7*a(n)+14*a(n+1)-13*a(n+2)+6*a(n+3), 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) = 19
a \! \left(n +4\right) = -7 a \! \left(n \right)+14 a \! \left(n +1\right)-13 a \! \left(n +2\right)+6 a \! \left(n +3\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18660/
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[18,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[12,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[17,x]
F[15,x] = 0
F[16,x] = F[10,x]*F[12,x]
F[17,x] = F[12,x]*F[13,x]
F[18,x] = F[19,x]+F[2,x]
F[19,x] = F[15,x]+F[20,x]+F[24,x]
F[20,x] = F[12,x]*F[21,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[6,x]
F[23,x] = F[19,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[33,x]
F[26,x] = F[2,x]+F[27,x]
F[27,x] = F[15,x]+F[28,x]+F[32,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[10,x]
F[31,x] = F[27,x]
F[32,x] = F[12,x]*F[26,x]
F[33,x] = F[34,x]+F[40,x]
F[34,x] = F[15,x]+F[35,x]+F[39,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[12,x]
F[38,x] = F[34,x]
F[39,x] = F[12,x]*F[2,x]
F[40,x] = F[15,x]+F[41,x]+F[42,x]+F[43,x]
F[41,x] = 0
F[42,x] = F[12,x]*F[27,x]
F[43,x] = F[12,x]*F[33,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_{18}\! \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_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{2}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{24}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{6}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{33}\! \left(x \right)
F_{26}\! \left(x \right) = F_{2}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{10}\! \left(x \right)
F_{31}\! \left(x \right) = F_{27}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{40}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right)
F_{38}\! \left(x \right) = F_{34}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{2}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right)+F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = 0
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \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_18(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_12(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_17(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_10(x)*F_12(x))
Eq(F_17(x), F_12(x)*F_13(x))
Eq(F_18(x), F_19(x) + F_2(x))
Eq(F_19(x), F_15(x) + F_20(x) + F_24(x))
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_6(x))
Eq(F_23(x), F_19(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_33(x))
Eq(F_26(x), F_2(x) + F_27(x))
Eq(F_27(x), F_15(x) + F_28(x) + F_32(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_10(x))
Eq(F_31(x), F_27(x))
Eq(F_32(x), F_12(x)*F_26(x))
Eq(F_33(x), F_34(x) + F_40(x))
Eq(F_34(x), F_15(x) + F_35(x) + F_39(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_12(x))
Eq(F_38(x), F_34(x))
Eq(F_39(x), F_12(x)*F_2(x))
Eq(F_40(x), F_15(x) + F_41(x) + F_42(x) + F_43(x))
Eq(F_41(x), 0)
Eq(F_42(x), F_12(x)*F_27(x))
Eq(F_43(x), F_12(x)*F_33(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, 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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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, 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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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": "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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 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