0123_0132_1032_1302_1320
Counting sequence:
1, 1, 2, 6, 19, 57, 164, 459, 1261, 3420, 9191, 24537, 65188, 172563, 455573, 1200300, 3157615, 8297121, 21782900, 57149883, 149863357, 392833404, 1029423287, 2697009321, 7064750404, 18503533347, 48458432549, 126896930124, 332282689471, 870051801585, 2278074041876, 5964572977227, 15616450196173, 40886388224028, 107045935701383, 280257861331065, 733740533193700, 1920989508053811, 5029279530575285, 13166952162887148, 34471783116516367, 90248809503522369, 236275470027771572, 618579249847234011, 1619465578048813789, 4239824081368974012, 11100019860197641559, 29060261887503017289, 76080818578869543556, 199182299402221879875, 521466290734028629061, 1365216995012329073292, 3574185538727888722783, 9357341310021197358993, 24497841769035423882068, 64136190752484515342955, 167910743999217004258285, 439596068266764261654876, 1150877514844271309152295, 3013036584352440722693913, 7888232454385832972713252, 20651661211150622423013459, 54066752043757162751462357, 141548596649503122741644076, 370579041363516719294010799, 970188534358576062781470177, 2539986575547269524332563444, 6649771219953348620780547579, 17409327139653008559137734141, 45578210309686141498889964540, 119325304010766344822046778871, 312397702165334750736279610857, 817867803370681622924850531268, 2141205709717597549114388938083, 5605749329323885886570550193253, 14676042285337609834901729462220, 38422377540856043066743573834495, 100591090365564718262546863323441, 263350893612506509515332758700180, 689461590585291605872322897905803, 1805033878370041899279378905274637, 4725640044978181274321299758432924, 12371886257471196288395492251053767, 32390018729248796320287120756787641, 84798169933901970131309757543427684, 222004491079710668991329926921732467, 581215303319737146678055573318243829, 1521641418908514990713587893225947244, 3983708953463836264804210306745494351, 10429485441599050682382047427782328705, 27304747371565429539707940778145077556, 71484756673561465451473792509740075547, 187149522650047421844177471957249492253, 489963811278437710139986693774357087548, 1282741911188979528693638750190519143063, 3358261922295928516176641838446595086985, 8792043855713661300307711328448055608580, 23017869644874765945689341273495150720131, 60261565078970057658646010745232554514565, 157766825592154249274020087468592828749068, 413038911697730374650957044673326563583647
Generating function in Maple syntax:
-(x^4+x^3+4*x^2-4*x+1)/(2*x-1)/(x^2-3*x+1)
Generating function in latex syntax:
-\frac{x^{4}+x^{3}+4 x^{2}-4 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(-x**4 - x**3 - 4*x**2 + 4*x - 1)/((2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*F(x)+x^4+x^3+4*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+x^{4}+x^{3}+4 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/20*(-12*5^(1/2)+40)*(3/2-1/2*5^(1/2))^(-n)+1/20*(12*5^(1/2)+40)*(3/2+1/2*5^(1/2))^(-n)-3/4*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-12 \sqrt{5}+40\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(12 \sqrt{5}+40\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}-\frac{3 \,2^{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(n+3) = 2*a(n)-7*a(n+1)+5*a(n+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) = 19
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16361/
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[19,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[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[12,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[62,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[28,x]
F[23,x] = F[24,x]+F[26,x]
F[24,x] = F[25,x]
F[25,x] = F[10,x]*F[4,x]
F[26,x] = F[13,x]+F[14,x]+F[27,x]
F[27,x] = F[11,x]*F[4,x]
F[28,x] = F[29,x]+F[53,x]
F[29,x] = F[13,x]+F[21,x]+F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[2,x]+F[32,x]
F[32,x] = F[13,x]+F[33,x]+F[52,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[12,x]+F[4,x]
F[36,x] = F[32,x]+F[37,x]
F[37,x] = F[13,x]+F[38,x]+F[49,x]+F[51,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[45,x]
F[40,x] = F[41,x]+F[43,x]
F[41,x] = F[42,x]
F[42,x] = F[16,x]*F[4,x]
F[43,x] = F[44,x]
F[44,x] = F[17,x]*F[4,x]
F[45,x] = F[46,x]
F[46,x] = 2*F[13,x]+F[38,x]+F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[32,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[48,x]
F[51,x] = 0
F[52,x] = F[2,x]*F[4,x]
F[53,x] = F[13,x]+F[49,x]+F[54,x]+F[61,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[59,x]
F[56,x] = F[57,x]+F[58,x]
F[57,x] = F[42,x]
F[58,x] = F[44,x]
F[59,x] = F[60,x]
F[60,x] = 2*F[13,x]+F[47,x]+F[54,x]
F[61,x] = 0
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[31,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_{19}\! \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_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{62}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{28}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{53}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right)+F_{33}\! \left(x \right)+F_{52}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right)+F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{32}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{13}\! \left(x \right)+F_{38}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{45}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{38}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{32}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = 0
F_{52}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{53}\! \left(x \right) = F_{13}\! \left(x \right)+F_{49}\! \left(x \right)+F_{54}\! \left(x \right)+F_{61}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{59}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{58}\! \left(x \right)
F_{57}\! \left(x \right) = F_{42}\! \left(x \right)
F_{58}\! \left(x \right) = F_{44}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{47}\! \left(x \right)+F_{54}\! \left(x \right)
F_{61}\! \left(x \right) = 0
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{31}\! \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_19(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_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_12(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_62(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_28(x))
Eq(F_23(x), F_24(x) + F_26(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_10(x)*F_4(x))
Eq(F_26(x), F_13(x) + F_14(x) + F_27(x))
Eq(F_27(x), F_11(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_53(x))
Eq(F_29(x), F_13(x) + F_21(x) + F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_2(x) + F_32(x))
Eq(F_32(x), F_13(x) + F_33(x) + F_52(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_12(x) + F_4(x))
Eq(F_36(x), F_32(x) + F_37(x))
Eq(F_37(x), F_13(x) + F_38(x) + F_49(x) + F_51(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_45(x))
Eq(F_40(x), F_41(x) + F_43(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_16(x)*F_4(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_17(x)*F_4(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), 2*F_13(x) + F_38(x) + F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_32(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_48(x))
Eq(F_51(x), 0)
Eq(F_52(x), F_2(x)*F_4(x))
Eq(F_53(x), F_13(x) + F_49(x) + F_54(x) + F_61(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_59(x))
Eq(F_56(x), F_57(x) + F_58(x))
Eq(F_57(x), F_42(x))
Eq(F_58(x), F_44(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), 2*F_13(x) + F_47(x) + F_54(x))
Eq(F_61(x), 0)
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_31(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": [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, 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": [1, 3, 2, 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