0123_0132_1032_1302_3102
Counting sequence:
1, 1, 2, 6, 19, 55, 152, 406, 1059, 2715, 6872, 17226, 42859, 106015, 261032, 640366, 1566339, 3822195, 9309032, 22636866, 54975979, 133375255, 323299352, 783119686, 1895830179, 4587362955, 11095721912, 26829138426, 64854662059, 156739789135, 378736893512, 915018882526, 2210385271299, 5339010650595, 12894849023432, 31141593599346, 75203806025899, 181600745258695, 438508375758392, 1058823655205686, 2556568003030179, 6172784294986875, 14903785860445592, 35983654550761386, 86877692031735019, 209752232753764735, 506408545818331112, 1222622100948560206, 2951758300831718019, 7126349808844529235, 17204880130985842472, 41536954495746346146, 100280477972338798699, 242101288140144471415, 584489809652068797272, 1411094418243164177446, 3406705667736161375139, 8224559796911015373675, 19855933347949249014392, 47936642665591627186266, 115729651024696730954539, 279396809406113544230575, 674524999219180729686152, 1628450266608988824143806, 3931432449966186019055619, 9491329001599416144418755, 22914118123281128872220552, 55319620588393895017514706, 133553469980533361164559659, 322426781910389501861253415, 778407476523170133916305272, 1879242620400445307752341526, 4536894488211491825537943459, 10953035138598291111062138715, 26442971848957798352130041432, 63838993003613336424257862666, 154120986190383368418517048939, 372081022052777867697034524895, 898283143632734692684071227432, 2168647535991838430808147237166, 5235578668963593909786306216579, 12639805780613390961352640699955, 30515192043579105254435349675752, 73670193494549060314110864169986, 177855586286232143570432126252779, 429381380574123182830525213149655, 1036618376448698179982582745500312, 2502618191499958884297891090046726, 6041854875505494631582765697386659, 14586328174624705513472224028405835, 35214511688982420390544816841369912, 85015352481044575758597063885488826, 205245218507981630835809356961033899, 495505793210827955286356602504929295, 1196256812357277777120804211365637832, 2888019432780663980952528324025695646, 6972295707629166681874987456996010499, 16832610907460119230400756433175679395, 40637517641391648914073006713663294792, 98107646427927904601339782641134119986, 236852810972616433202338597557195236779
Generating function in Maple syntax:
(2*x^5-2*x^3-4*x^2+4*x-1)/(x-1)/(2*x-1)/(x^2+2*x-1)
Generating function in latex syntax:
\frac{2 x^{5}-2 x^{3}-4 x^{2}+4 x -1}{\left(x -1\right) \left(2 x -1\right) \left(x^{2}+2 x -1\right)}
Generating function in sympy syntax:
(2*x**5 - 2*x**3 - 4*x**2 + 4*x - 1)/((x - 1)*(2*x - 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)*(2*x-1)*(x^2+2*x-1)*F(x)+2*x^5-2*x^3-4*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(2 x -1\right) \left(x^{2}+2 x -1\right) F \! \left(x \right)+2 x^{5}-2 x^{3}-4 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/2+5/4*(-1-2^(1/2))^(-n)-3*2^(-1+n)+5/4*(2^(1/2)-1)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{1}{2}+\frac{5 \left(-1-\sqrt{2}\right)^{-n}}{4}-3 \,2^{-1+n}+\frac{5 \left(\sqrt{2}-1\right)^{-n}}{4} & \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(5) = 55
a(n+3) = -2*a(n)-3*a(n+1)+4*a(n+2)+1, n >= 6
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) = 55
a \! \left(n +3\right) = -2 a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right)+1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16368/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[24,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[18,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[10,x]+F[15,x]
F[15,x] = F[12,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[12,x]*F[4,x]
F[18,x] = F[17,x]+F[19,x]+F[20,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[4,x]
F[23,x] = F[18,x]
F[24,x] = F[25,x]+F[39,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[1,x]+F[12,x]
F[29,x] = F[25,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[25,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[25,x]*F[4,x]
F[36,x] = F[30,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[30,x]*F[4,x]
F[39,x] = F[19,x]+F[40,x]+F[64,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[47,x]
F[42,x] = F[43,x]+F[45,x]
F[43,x] = F[44,x]
F[44,x] = F[10,x]*F[4,x]
F[45,x] = F[19,x]+F[20,x]+F[46,x]
F[46,x] = F[11,x]*F[4,x]
F[47,x] = F[48,x]+F[61,x]
F[48,x] = F[19,x]+F[40,x]+F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[25,x]+F[51,x]
F[51,x] = F[19,x]+F[35,x]+F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[18,x]+F[4,x]
F[55,x] = F[51,x]+F[56,x]
F[56,x] = 2*F[19,x]+F[38,x]+F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[51,x]
F[60,x] = F[56,x]
F[61,x] = 2*F[19,x]+F[57,x]+F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[30,x]+F[56,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[50,x]+F[63,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{24}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \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_{10}\! \left(x \right)+F_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{17}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{18}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{39}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{25}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{30}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{19}\! \left(x \right)+F_{40}\! \left(x \right)+F_{64}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{47}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{45}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{61}\! \left(x \right)
F_{48}\! \left(x \right) = F_{19}\! \left(x \right)+F_{40}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{25}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{19}\! \left(x \right)+F_{35}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{55}\! \left(x \right) = F_{51}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{38}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{51}\! \left(x \right)
F_{60}\! \left(x \right) = F_{56}\! \left(x \right)
F_{61}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{57}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{30}\! \left(x \right)+F_{56}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{50}\! \left(x \right)+F_{63}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_24(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_18(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_10(x) + F_15(x))
Eq(F_15(x), F_12(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_12(x)*F_4(x))
Eq(F_18(x), F_17(x) + F_19(x) + F_20(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_4(x))
Eq(F_23(x), F_18(x))
Eq(F_24(x), F_25(x) + F_39(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_1(x) + F_12(x))
Eq(F_29(x), F_25(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_25(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_25(x)*F_4(x))
Eq(F_36(x), F_30(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_30(x)*F_4(x))
Eq(F_39(x), F_19(x) + F_40(x) + F_64(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_47(x))
Eq(F_42(x), F_43(x) + F_45(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_10(x)*F_4(x))
Eq(F_45(x), F_19(x) + F_20(x) + F_46(x))
Eq(F_46(x), F_11(x)*F_4(x))
Eq(F_47(x), F_48(x) + F_61(x))
Eq(F_48(x), F_19(x) + F_40(x) + F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_25(x) + F_51(x))
Eq(F_51(x), F_19(x) + F_35(x) + F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_18(x) + F_4(x))
Eq(F_55(x), F_51(x) + F_56(x))
Eq(F_56(x), 2*F_19(x) + F_38(x) + F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_51(x))
Eq(F_60(x), F_56(x))
Eq(F_61(x), 2*F_19(x) + F_57(x) + F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_30(x) + F_56(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_50(x) + F_63(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, 1, 3, 2], "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, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 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, 1, 3, 2], "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, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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