0213_0231_1230_3102
Counting sequence:
1, 1, 2, 6, 20, 63, 181, 482, 1213, 2929, 6860, 15703, 35321, 78370, 172009, 374229, 808324, 1735451, 3706973, 7883570, 16702277, 35268137, 74252572, 155920111, 326641585, 682839618, 1424717201, 2967389037, 6170491028, 12812090259, 26565882805, 55014338258, 113792475661, 235110371425, 485268058604, 1000625045959, 2061418722089, 4243159774306, 8726940051193, 17935082019525, 36832504627492, 75589588097867, 155028168301517, 317754052900466, 650903104901525, 1332595506595673, 2726768471873596, 5576690024799967, 11399683240490593, 23291968055235714, 47569131480238625, 97108641113742813, 198158018169005876, 404197475269772355, 824157775086774949, 1679841113000439314, 3422733212070795229, 6971568170429990161, 14195339468001483788, 28895084598999244855, 58798979567269018457, 119615578325070338722, 243266392526474499337, 494603252752877104821, 1005347434348140102340, 2042976715770842132603, 4150517108523127943741, 8430161543231253209522, 17118577693886930850533, 34753664529899250316169, 70540347226380247401820, 143146730595531497633743, 290425533168543479757841, 589115209793593916617410, 1194758705693686214389169, 2422573982295399646158669, 4911261104295368649100244, 9954748484583421388859891, 20173949515624271258154133, 40876804115219005153385810, 82811418383906601556250797, 167738457051333657262992577, 339708154631819160453823532, 687878790260636222041712551, 1392682542416073393256800905, 2819215008461248041043986658, 5706129863921003094237620761, 11547659421418824072047047077, 23366118229311392273599093348, 47273835230470185027842083499, 95630868002855191600967246765, 193428131086659959098129837298, 391189052330559023378274832181, 791043684968056143315833633465, 1599418530537788319335113328764, 3233499382259188429857250558399, 6536323406853660007453558819009, 13211296098326205601530374719746, 26699890765806561232817415180353, 53954378669786120672803455096765, 109017951615699314764109604497012
Generating function in Maple syntax:
(x^7+4*x^6-4*x^5+2*x^4+9*x^3-13*x^2+6*x-1)/(x^2+x-1)/(2*x-1)^2/(-1+x)^2
Generating function in latex syntax:
\frac{x^{7}+4 x^{6}-4 x^{5}+2 x^{4}+9 x^{3}-13 x^{2}+6 x -1}{\left(x^{2}+x -1\right) \left(2 x -1\right)^{2} \left(-1+x \right)^{2}}
Generating function in sympy syntax:
(x**7 + 4*x**6 - 4*x**5 + 2*x**4 + 9*x**3 - 13*x**2 + 6*x - 1)/((x - 1)**2*(2*x - 1)**2*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2+x-1)*(2*x-1)^2*(-1+x)^2*F(x)+x^7+4*x^6-4*x^5+2*x^4+9*x^3-13*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}+x -1\right) \left(2 x -1\right)^{2} \left(-1+x \right)^{2} F \! \left(x \right)+x^{7}+4 x^{6}-4 x^{5}+2 x^{4}+9 x^{3}-13 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/40*(44*5^(1/2)-100)*(-1/2*5^(1/2)-1/2)^(-n)+1/40*(-44*5^(1/2)-100)*(1/2*5^(1/2)-1/2)^(-n)+7/8*2^n*n+4*n-3*2^(-1+n)+6)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(44 \sqrt{5}-100\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{40}+\frac{\left(-44 \sqrt{5}-100\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{40}+\frac{7 \,2^{n} n}{8}\\+4 n -3 \,2^{-1+n}+6 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 63
a(6) = 181
a(7) = 482
a(n) = 7/4*a(n+2)-5/4*a(n+3)+1/4*a(n+4)+n-3/2, 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) = 20
a \! \left(5\right) = 63
a \! \left(6\right) = 181
a \! \left(7\right) = 482
a \! \left(n \right) = \frac{7 a \! \left(n +2\right)}{4}-\frac{5 a \! \left(n +3\right)}{4}+\frac{a \! \left(n +4\right)}{4}+n -\frac{3}{2}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15703/
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[11,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[6,x]
F[10,x] = F[7,x]
F[11,x] = F[12,x]+F[21,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[15,x]+F[6,x]
F[15,x] = F[12,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[20,x]
F[19,x] = F[12,x]
F[20,x] = F[16,x]
F[21,x] = F[22,x]+F[23,x]+F[54,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[32,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[22,x]+F[27,x]+F[29,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[25,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[10,x]+F[31,x]
F[31,x] = F[26,x]
F[32,x] = F[21,x]+F[33,x]
F[33,x] = 2*F[22,x]+F[34,x]+F[47,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[43,x]
F[39,x] = F[40,x]+F[7,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[39,x]
F[43,x] = F[36,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[36,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[50,x]
F[49,x] = F[16,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[20,x]+F[53,x]
F[53,x] = F[51,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[12,x]+F[36,x]
F[57,x] = F[58,x]
F[58,x] = F[22,x]+F[59,x]+F[66,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[25,x]+F[61,x]
F[61,x] = F[16,x]+F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[20,x]+F[65,x]
F[65,x] = F[62,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[19,x]+F[68,x]
F[68,x] = F[58,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_{11}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{7}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{21}\! \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_{15}\! \left(x \right)+F_{6}\! \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_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{54}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{29}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{25}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{10}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{26}\! \left(x \right)
F_{32}\! \left(x \right) = F_{21}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{34}\! \left(x \right)+F_{47}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \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_{39}\! \left(x \right)+F_{43}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{7}\! \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_{39}\! \left(x \right)
F_{43}\! \left(x \right) = F_{36}\! \left(x \right)+F_{44}\! \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_{36}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{16}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{20}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{51}\! \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_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right)+F_{36}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{22}\! \left(x \right)+F_{59}\! \left(x \right)+F_{66}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{25}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{16}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{20}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{62}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{19}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{58}\! \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_11(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_6(x))
Eq(F_10(x), F_7(x))
Eq(F_11(x), F_12(x) + F_21(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_6(x))
Eq(F_15(x), F_12(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_20(x))
Eq(F_19(x), F_12(x))
Eq(F_20(x), F_16(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_54(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_32(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_22(x) + F_27(x) + F_29(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_25(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_10(x) + F_31(x))
Eq(F_31(x), F_26(x))
Eq(F_32(x), F_21(x) + F_33(x))
Eq(F_33(x), 2*F_22(x) + F_34(x) + F_47(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(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_39(x) + F_43(x))
Eq(F_39(x), F_40(x) + F_7(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_39(x))
Eq(F_43(x), F_36(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_36(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_50(x))
Eq(F_49(x), F_16(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_20(x) + F_53(x))
Eq(F_53(x), F_51(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_12(x) + F_36(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_22(x) + F_59(x) + F_66(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_25(x) + F_61(x))
Eq(F_61(x), F_16(x) + F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_20(x) + F_65(x))
Eq(F_65(x), F_62(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_19(x) + F_68(x))
Eq(F_68(x), F_58(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, 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, 2, 3, 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, 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, 2, 3, 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": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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