0132_0231_1023_1320_2031
Counting sequence:
1, 1, 2, 6, 19, 54, 145, 380, 989, 2574, 6711, 17526, 45821, 119876, 313729, 821214, 2149795, 5628030, 14734129, 38574164, 100988141, 264390006, 692181591, 1812154446, 4744281389, 12420689324, 32517786145, 85132668630, 222880219219, 583507988454, 1527643745521, 3999423247436, 10470625996061, 27412454739966, 71766738222999, 187887759928134, 491896541560445, 1287801864752180, 3371509052695009, 8826725293331694, 23108666827298851, 60499275188563566, 158389158738390481, 414668201026606436, 1085615444341427309, 2842178131997673894, 7440918951651592695, 19480578722957102430, 51000817217219712749, 133521872928702033884, 349564801568886386881, 915172531777957124646, 2395952793764984984851, 6272685849516997827606, 16422104754786008495569, 42993628414841027656604, 112558780489737074471645, 294682713054370195755630, 771489358673373512792439, 2019785362965750342618774, 5287866730223877515060861, 13843814827705882202560676, 36243577752893769092617921, 94886918430975425075289726, 248417177540032506133247779, 650364614189122093324450014, 1702676665027333773840098545, 4457665380892879228195841780, 11670319477651303910747422829, 30553293052061032504046422614, 79989559678531793601391840791, 209415385983534348300129095406, 548256598272071251298995440941, 1435354408832679405596857222796, 3757806628225966965491576222689, 9838065475845221490877871440374, 25756389799309697507142038093395, 67431103922083871030548242834630, 176536921966941915584502690405169, 462179661978741875722959828375404, 1210002063969283711584376794715421, 3167826529929109259030170555765086, 8293477525818044065506134872573911, 21712606047525022937488234061950566, 56844340616757024746958567313271549, 148820415802746051303387467877857684, 389616906791481129163203836320294945, 1020030304571697336186224041083020430, 2670474006923610879395468286928759459, 6991391716199135302000180819703250894, 18303701141673795026605074172180986001, 47919711708822249777815041696839699716, 125455433984792954306840050918338105581, 328446590245556613142705111058174609286, 859884336751876885121275282256185714359, 2251206420010074042221120735710382525694, 5893734923278345241542086924874961854445, 15429998349824961682405140038914503029180, 40396260126196539805673333191868547224449, 105758782028764657734614859536691138635334, 276880085960097433398171245418204868672531
Generating function in Maple syntax:
(x^7-x^6+3*x^5-2*x^4+6*x^3-9*x^2+5*x-1)/(x^2-3*x+1)/(x-1)^3
Generating function in latex syntax:
\frac{x^{7}-x^{6}+3 x^{5}-2 x^{4}+6 x^{3}-9 x^{2}+5 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**7 - x**6 + 3*x**5 - 2*x**4 + 6*x**3 - 9*x**2 + 5*x - 1)/((x - 1)**3*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(x^2-3*x+1)*(x-1)^3*F(x)+x^7-x^6+3*x^5-2*x^4+6*x^3-9*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{7}-x^{6}+3 x^{5}-2 x^{4}+6 x^{3}-9 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/5*(39*5^(1/2)-85)*(3/2-1/2*5^(1/2))^(-n)+1/5*(-39*5^(1/2)-85)*(3/2+1/2*5^(1/2))^(-n)+n^2-8*n+15)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(39 \sqrt{5}-85\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{5}+\frac{\left(-39 \sqrt{5}-85\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{5}+n^{2}-8 n +15 & \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) = 54
a(6) = 145
a(7) = 380
a(n+2) = -n^2-a(n)+3*a(n+1)+6*n-6, n >= 8
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) = 54
a \! \left(6\right) = 145
a \! \left(7\right) = 380
a \! \left(n +2\right) = -n^{2}-a \! \left(n \right)+3 a \! \left(n +1\right)+6 n -6, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18791/
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[22,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[20,x]
F[17,x] = F[18,x]+F[4,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[15,x]
F[22,x] = F[2,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[64,x]
F[24,x] = 0
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[32,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[1,x]+F[28,x]
F[31,x] = F[15,x]
F[32,x] = F[33,x]+F[52,x]
F[33,x] = F[24,x]+F[25,x]+F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[2,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[27,x]+F[39,x]
F[39,x] = F[36,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[44,x]+F[50,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[21,x]+F[4,x]
F[48,x] = F[44,x]+F[49,x]
F[49,x] = F[41,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[43,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[56,x]+F[62,x]
F[56,x] = F[24,x]+F[57,x]+F[61,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[47,x]+F[59,x]
F[59,x] = F[56,x]+F[60,x]
F[60,x] = F[53,x]
F[61,x] = F[2,x]*F[4,x]
F[62,x] = F[63,x]
F[63,x] = F[4,x]*F[55,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[69,x]
F[66,x] = F[2,x]+F[67,x]
F[67,x] = F[24,x]+F[57,x]+F[68,x]
F[68,x] = F[4,x]*F[66,x]
F[69,x] = F[37,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_{22}\! \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_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{20}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{64}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{32}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{15}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{52}\! \left(x \right)
F_{33}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{2}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{27}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{50}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{21}\! \left(x \right)+F_{4}\! \left(x \right)
F_{48}\! \left(x \right) = F_{44}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{41}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{62}\! \left(x \right)
F_{56}\! \left(x \right) = F_{24}\! \left(x \right)+F_{57}\! \left(x \right)+F_{61}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{47}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{56}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{53}\! \left(x \right)
F_{61}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{69}\! \left(x \right)
F_{66}\! \left(x \right) = F_{2}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{24}\! \left(x \right)+F_{57}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{69}\! \left(x \right) = F_{37}\! \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_22(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_20(x))
Eq(F_17(x), F_18(x) + F_4(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_15(x))
Eq(F_22(x), F_2(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_64(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_32(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_1(x) + F_28(x))
Eq(F_31(x), F_15(x))
Eq(F_32(x), F_33(x) + F_52(x))
Eq(F_33(x), F_24(x) + F_25(x) + F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_2(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_27(x) + F_39(x))
Eq(F_39(x), F_36(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_44(x) + F_50(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_21(x) + F_4(x))
Eq(F_48(x), F_44(x) + F_49(x))
Eq(F_49(x), F_41(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_43(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_56(x) + F_62(x))
Eq(F_56(x), F_24(x) + F_57(x) + F_61(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_47(x) + F_59(x))
Eq(F_59(x), F_56(x) + F_60(x))
Eq(F_60(x), F_53(x))
Eq(F_61(x), F_2(x)*F_4(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_4(x)*F_55(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_69(x))
Eq(F_66(x), F_2(x) + F_67(x))
Eq(F_67(x), F_24(x) + F_57(x) + F_68(x))
Eq(F_68(x), F_4(x)*F_66(x))
Eq(F_69(x), F_37(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": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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