0132_0213_0231_1320_3102
Counting sequence:
1, 1, 2, 6, 19, 53, 135, 336, 842, 2140, 5505, 14275, 37189, 97126, 253984, 664578, 1739455, 4553441, 11920467, 31207500, 81701510, 213896440, 559987149, 1466064271, 3838204849, 10048549378, 26307442300, 68873776446, 180313885867, 472067879885, 1235889752415, 3235601375880, 8470914373634, 22177141743316, 58060510854489, 152004390818203, 397952661598045, 1041853593973726, 2727608120320792, 7140970766986170, 18695304180635095, 48944941774916345, 128139521144111019, 335473621657413636, 878281343828126654, 2299370409826962928, 6019829885652758565, 15760119247131309031, 41260527855741164617, 108021464320092180730, 282803865104535373300, 740390130993513934710, 1938366527876006426179, 5074709452634505338981, 13285761830027509585719, 34782576037448023412928, 91061966282316560647610, 238403322809501658524236, 624148002146188414919217, 1634040683629063586227315, 4277974048741002343756405, 11199881462593943445035350, 29321670339040827991342864, 76765129554528540528986226, 200973718324544793595608559, 526156025419105840257831953, 1377494357932772727177879555, 3606327048379212341275798716, 9441486787204864296649508342, 24718133313235380548672717800, 64712913152501277349368636285, 169420606144268451499433182015, 443548905280304077148930900449, 1161226109696643779947359509746, 3040129423809627262693147618924, 7959162161732238008132083336878, 20837357061387086761703102381275, 54552909022429022276977223796221, 142821370005899980069228568996367, 373911200995270917930708483181560, 978912232979912773722896880536690, 2562825497944467403237982158416580, 6709564260853489435991049594700809, 17565867284616000904735166625673291, 45988037592994513278214450282306189, 120398245494367538929908184221232078, 315206698890108103511510102381376520, 825221851175956771604622122922883626, 2160458854637762211302356266387260167, 5656154712737329862302446676238882345, 14808005283574227375604983762329371995, 38767861137985352264512504610749218420, 101495578130381829417932530069918267694, 265718873253160135989285085599005568736, 695661041629098578549922726727098422229, 1821264251634135599660483094582289681303, 4768131713273308220431526557019770604665, 12483130888185789061634096576477022115306, 32681260951284058964470763172411295723492, 85560651965666387831778192940756865037030, 224000694945715104530863815649859299369075
Generating function in Maple syntax:
-(2*x^7-3*x^6-4*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{2 x^{7}-3 x^{6}-4 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:
(-2*x**7 + 3*x**6 + 4*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)+2*x^7-3*x^6-4*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)+2 x^{7}-3 x^{6}-4 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 = 2,2,1/10*(-23*5^(1/2)+55)*(3/2-1/2*5^(1/2))^(-n)+1/10*(23*5^(1/2)+55)*(3/2+1/2*5^(1/2))^(-n)+2*n^2-11*n+14)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-23 \sqrt{5}+55\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(23 \sqrt{5}+55\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+2 n^{2}-11 n \\+14 & \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) = 53
a(6) = 135
a(7) = 336
a(n+2) = -2*n^2-a(n)+3*a(n+1)+7*n-1, 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) = 53
a \! \left(6\right) = 135
a \! \left(7\right) = 336
a \! \left(n +2\right) = -2 n^{2}-a \! \left(n \right)+3 a \! \left(n +1\right)+7 n -1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18183/
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[14,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[7,x]
F[14,x] = F[15,x]+F[25,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[6,x]
F[18,x] = F[15,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]
F[22,x] = F[15,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[22,x]*F[4,x]
F[25,x] = F[26,x]+F[27,x]+F[57,x]
F[26,x] = 0
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[40,x]
F[29,x] = F[30,x]+F[7,x]
F[30,x] = F[26,x]+F[31,x]+F[36,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[11,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[33,x]*F[4,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[13,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[31,x]
F[40,x] = F[25,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[45,x]+F[55,x]
F[45,x] = F[26,x]+F[46,x]+F[53,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[11,x]+F[49,x]
F[49,x] = F[26,x]+F[31,x]+F[50,x]
F[50,x] = F[13,x]*F[4,x]
F[51,x] = F[45,x]+F[52,x]
F[52,x] = F[42,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[15,x]+F[45,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[44,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[54,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[39,x]+F[7,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_{14}\! \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_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{6}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
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_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{57}\! \left(x \right)
F_{26}\! \left(x \right) = 0
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{40}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{7}\! \left(x \right)
F_{30}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \left(x \right)+F_{36}\! \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_{33}\! \left(x \right) = F_{11}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{13}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{31}\! \left(x \right)
F_{40}\! \left(x \right) = F_{25}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{55}\! \left(x \right)
F_{45}\! \left(x \right) = F_{26}\! \left(x \right)+F_{46}\! \left(x \right)+F_{53}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{11}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{51}\! \left(x \right) = F_{45}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{42}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{15}\! \left(x \right)+F_{45}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{54}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{39}\! \left(x \right)+F_{7}\! \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_14(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_7(x))
Eq(F_14(x), F_15(x) + F_25(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_6(x))
Eq(F_18(x), F_15(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_15(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_22(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_27(x) + F_57(x))
Eq(F_26(x), 0)
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_40(x))
Eq(F_29(x), F_30(x) + F_7(x))
Eq(F_30(x), F_26(x) + F_31(x) + F_36(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_11(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_33(x)*F_4(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_13(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_31(x))
Eq(F_40(x), F_25(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_45(x) + F_55(x))
Eq(F_45(x), F_26(x) + F_46(x) + F_53(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_11(x) + F_49(x))
Eq(F_49(x), F_26(x) + F_31(x) + F_50(x))
Eq(F_50(x), F_13(x)*F_4(x))
Eq(F_51(x), F_45(x) + F_52(x))
Eq(F_52(x), F_42(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_15(x) + F_45(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_44(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_54(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_39(x) + F_7(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, 1, 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": [1, 3, 2, 0], "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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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