0123_0231_0312_1230_1302
Counting sequence:
1, 1, 2, 6, 19, 59, 179, 533, 1564, 4536, 13031, 37141, 105157, 296041, 829334, 2313354, 6428539, 17804303, 49162679, 135386621, 371927536, 1019484156, 2788859087, 7615007401, 20757571849, 56494020049, 153532015274, 416688294798, 1129484149219, 3058031724131, 8270462456891, 22344642376421, 60311473428292, 162642517446336, 438226288768439, 1179814238894269, 3173939888162509, 8532397916302393, 22921707872623934, 61537695246498066, 165107832489776971, 442730196546624311, 1186499485498563647, 3178074050670678221, 8508223310329838104, 22766792021046325764, 60892240531175241119, 162789996766850213713, 435017863574121746449, 1162003868175383248801, 3102674449806886323026, 8281321333589982466326, 22095514399142322991027, 58932594556030065505931, 157130162497347848609219, 418814199929020326569909, 1115953465040333694762604, 2972612971450835602456008, 7915944750338117083157639, 21073832406382492713935461, 56087326548240348288852181, 149234858349812536776312841, 396975607756188228680958182, 1055720331572251064565485466, 2806912127666070744271397659, 7461167906888979590717482079, 19828320418684417516031472839, 52682827970751902999359434653, 139945538532650632119843901504, 371670878122850385231580982796, 986892992283772358551328115311, 2619968697576431803480281855481, 6954069000541546556086723859737, 18454445405488313264313630457201, 48964832620147685931258183303674, 129894541566187517213139226095774, 344526694007888891064790399117699, 913654757134668459367446377017523, 2422537325498210371839561045081419, 6422287246989055007171059291906181, 17023214750254137817931072118528916, 45115697980499815600374700393676496, 119550011786639497276193203764596567, 316744394188874783953452891676405261, 839087208612958989466909238891399389, 2222519288339468481370258147101841369, 5886072788640871210530071401099489934, 15586503417599953120955591330148675618, 41268248351973987308657402212517313259, 109251869961750198303243078902963545991, 289193435615746176939430525659883393871, 765413030809468851031898107971656561261, 2025593364502132362368453936776485997672, 5359925591841363676193184508027030629396, 14181311290765793000358071865790893775679, 37516833390543082887202227119135285155489, 99240536331380824014065225302497577179169, 262485992845064426651122100191215658389313, 694189746477690200074870413668841418522274, 1835718942168174368484068504605526446770726, 4853901862493459745973503853120699450036051
Generating function in Maple syntax:
-(x^3-7*x^2+5*x-1)/(x^2-3*x+1)^2
Generating function in latex syntax:
-\frac{x^{3}-7 x^{2}+5 x -1}{\left(x^{2}-3 x +1\right)^{2}}
Generating function in sympy syntax:
(-x**3 + 7*x**2 - 5*x + 1)/(x**2 - 3*x + 1)**2
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)^2*F(x)+x^3-7*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)+x^{3}-7 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
1/10*(((-3/2*n-13/5)*5^(1/2)+7/2*n+6)*(3/2-1/2*5^(1/2))^(-n)+1/10*(5^(1/2)+10*n+15)*(3/2+1/2*5^(1/2))^(-n))*(3+5^(1/2))
Explicit closed form in latex syntax:
\frac{\left(\left(\left(-\frac{3 n}{2}-\frac{13}{5}\right) \sqrt{5}+\frac{7 n}{2}+6\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}+\frac{\left(\sqrt{5}+10 n +15\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}\right) \left(3+\sqrt{5}\right)}{10}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = -a(n)+6*a(n+1)-11*a(n+2)+6*a(n+3), n >= 4
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(n +4\right) = -a \! \left(n \right)+6 a \! \left(n +1\right)-11 a \! \left(n +2\right)+6 a \! \left(n +3\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17024/
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[17,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[10,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[10,x]+F[14,x]
F[17,x] = F[18,x]+F[29,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[21,x]+F[9,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]+F[24,x]+F[28,x]
F[23,x] = 0
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[10,x]
F[27,x] = F[22,x]
F[28,x] = F[12,x]*F[21,x]
F[29,x] = F[23,x]+F[30,x]+F[50,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[34,x]
F[32,x] = F[33,x]+F[6,x]
F[33,x] = F[15,x]
F[34,x] = F[29,x]+F[35,x]
F[35,x] = 2*F[23,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[33,x]
F[39,x] = F[35,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[23,x]+F[28,x]+F[43,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[16,x]+F[41,x]
F[45,x] = 2*F[23,x]+F[40,x]+F[46,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[14,x]
F[49,x] = F[45,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[18,x]+F[42,x]
F[53,x] = F[22,x]+F[45,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_{17}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{29}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{9}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{23}\! \left(x \right) = 0
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_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{10}\! \left(x \right)
F_{27}\! \left(x \right) = F_{22}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{29}\! \left(x \right) = F_{23}\! \left(x \right)+F_{30}\! \left(x \right)+F_{50}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{6}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)
F_{34}\! \left(x \right) = F_{29}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = 2 F_{23}\! \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_{33}\! \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_{45}\! \left(x \right)
F_{42}\! \left(x \right) = F_{23}\! \left(x \right)+F_{28}\! \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_{16}\! \left(x \right)+F_{41}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{40}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{14}\! \left(x \right)
F_{49}\! \left(x \right) = F_{45}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{18}\! \left(x \right)+F_{42}\! \left(x \right)
F_{53}\! \left(x \right) = F_{22}\! \left(x \right)+F_{45}\! \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_17(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_10(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_10(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_29(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_9(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_28(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_10(x))
Eq(F_27(x), F_22(x))
Eq(F_28(x), F_12(x)*F_21(x))
Eq(F_29(x), F_23(x) + F_30(x) + F_50(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_34(x))
Eq(F_32(x), F_33(x) + F_6(x))
Eq(F_33(x), F_15(x))
Eq(F_34(x), F_29(x) + F_35(x))
Eq(F_35(x), 2*F_23(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_33(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) + F_45(x))
Eq(F_42(x), F_23(x) + F_28(x) + F_43(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_16(x) + F_41(x))
Eq(F_45(x), 2*F_23(x) + F_40(x) + F_46(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_14(x))
Eq(F_49(x), F_45(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_18(x) + F_42(x))
Eq(F_53(x), F_22(x) + F_45(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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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, 0, 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