0132_1023_1032_1230_3021
Counting sequence:
1, 1, 2, 6, 19, 51, 122, 274, 592, 1248, 2589, 5314, 10829, 21959, 44375, 89453, 180000, 361719, 726160, 1456656, 2920250, 5851638, 11721199, 23471288, 46989199, 94053701, 188229097, 376654939, 753628042, 1507770693, 3016373834, 6034094374, 12070367524, 24144260124, 48294223665, 96597675358, 193210281665, 386444721779, 772928532395, 1545920311481, 3091942957860, 6184051496643, 12368370908404, 24737175312108, 49475052033854, 98951238971826, 197904314256547, 395811599729204, 791628006986467, 1583263792716113, 3166540171702429, 6333100708417159, 12666234368115694, 25332522052523889, 50665130372620430, 101330400329104738, 202661026509644680, 405322418454587064, 810645428195905605, 1621291813113838906, 3242585174236435445, 6485172853203654623, 12970349759146848959, 25940706075764019653, 51881422761737898984, 103762862691155977551, 207525753160201992472, 415051551265974199944, 830103175255408650242, 1660206468179847763758, 3320413126752186238999, 6640826561565893650544, 13281653621585799182839, 26563308049687131417629, 53126617404343807766545, 106253236920172693513843, 212506477256800009937170, 425012960041539720761949, 850025929027473765318290, 1700051872527281555315806, 3400103768471291459102380, 6800207574831645291351828, 13600415210969081304318489, 27200830521133015703395798, 54401661202766675223162089, 108803322665228847357450251, 217606645750653835442393747, 435213292181199308523403409, 870426585462486395412912492, 1740853172704952206830543003, 3481706348289971608031906044, 6963412701239989826439346404, 13926825410020093457625043334, 27853650832240347330371967594, 55707301684220968880612162635, 111414603400382372396214429548, 222829206852445453647287186635, 445658413788512050784422800809, 891316827712325953913552352325, 1782633655643574903661659878431, 3565267311641374655502581676790
Generating function in Maple syntax:
(x^9+x^8-4*x^7+2*x^5+2*x^4+4*x^3-9*x^2+5*x-1)/(2*x-1)/(x^2+x-1)/(x-1)^3
Generating function in latex syntax:
\frac{x^{9}+x^{8}-4 x^{7}+2 x^{5}+2 x^{4}+4 x^{3}-9 x^{2}+5 x -1}{\left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**9 + x**8 - 4*x**7 + 2*x**5 + 2*x**4 + 4*x**3 - 9*x**2 + 5*x - 1)/((x - 1)**3*(2*x - 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x^2+x-1)*(x-1)^3*F(x)+x^9+x^8-4*x^7+2*x^5+2*x^4+4*x^3-9*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{9}+x^{8}-4 x^{7}+2 x^{5}+2 x^{4}+4 x^{3}-9 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
-1/2*n^2+1/2*piecewise(n = 0,51/8,n = 1,19/4,n = 2,7/2,n = 3,1,0)-1/2*n-3+2/5*(-1/2*5^(1/2)-1/2)^(-n)*5^(1/2)-2/5*(1/2*5^(1/2)-1/2)^(-n)*5^(1/2)-(-1/2*5^(1/2)-1/2)^(-n)-(1/2*5^(1/2)-1/2)^(-n)+45/16*2^n
Explicit closed form in latex syntax:
-\frac{n^{2}}{2}+\frac{\left(\left\{\begin{array}{cc}\frac{51}{8} & n =0 \\ \frac{19}{4} & n =1 \\ \frac{7}{2} & n =2 \\ 1 & n =3 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)}{2}-\frac{n}{2}-3+\frac{2 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}-\frac{2 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \sqrt{5}}{5}-\left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}-\left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}+\frac{45 \,2^{n}}{16}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 51
a(6) = 122
a(7) = 274
a(8) = 592
a(9) = 1248
a(n+3) = -2*a(n)-a(n+1)+3*a(n+2)-1/2*(-2+n)*(-1+n), n >= 10
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) = 51
a \! \left(6\right) = 122
a \! \left(7\right) = 274
a \! \left(8\right) = 592
a \! \left(9\right) = 1248
a \! \left(n +3\right) = -2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)-\frac{\left(-2+n \right) \left(-1+n \right)}{2}, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19578/
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[35,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[16,x]+F[32,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]+F[4,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[4,x]
F[21,x] = F[22,x]+F[24,x]
F[22,x] = F[15,x]+F[16,x]+F[23,x]
F[23,x] = F[4,x]*F[7,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[18,x]+F[30,x]
F[30,x] = F[27,x]+F[31,x]
F[31,x] = F[25,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[7,x]
F[34,x] = F[28,x]
F[35,x] = F[36,x]+F[7,x]
F[36,x] = F[15,x]+F[37,x]+F[47,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[4,x]+F[40,x]
F[40,x] = F[15,x]+F[20,x]+F[41,x]
F[41,x] = F[11,x]*F[4,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[15,x]+F[23,x]+F[37,x]
F[44,x] = F[15,x]+F[25,x]+F[45,x]+F[46,x]
F[45,x] = 0
F[46,x] = F[14,x]*F[4,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[33,x]+F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[15,x]+F[16,x]+F[47,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[26,x]+F[54,x]
F[54,x] = F[52,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_{35}\! \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_{16}\! \left(x \right)+F_{32}\! \left(x \right)
F_{15}\! \left(x \right) = 0
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_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{24}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
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_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{7}\! \left(x \right)
F_{34}\! \left(x \right) = F_{28}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{7}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{47}\! \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_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{37}\! \left(x \right)
F_{44}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = 0
F_{46}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{33}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{47}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{26}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{52}\! \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_35(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) + F_16(x) + F_32(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x) + F_4(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_24(x))
Eq(F_22(x), F_15(x) + F_16(x) + F_23(x))
Eq(F_23(x), F_4(x)*F_7(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_18(x) + F_30(x))
Eq(F_30(x), F_27(x) + F_31(x))
Eq(F_31(x), F_25(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_7(x))
Eq(F_34(x), F_28(x))
Eq(F_35(x), F_36(x) + F_7(x))
Eq(F_36(x), F_15(x) + F_37(x) + F_47(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_4(x) + F_40(x))
Eq(F_40(x), F_15(x) + F_20(x) + F_41(x))
Eq(F_41(x), F_11(x)*F_4(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_15(x) + F_23(x) + F_37(x))
Eq(F_44(x), F_15(x) + F_25(x) + F_45(x) + F_46(x))
Eq(F_45(x), 0)
Eq(F_46(x), F_14(x)*F_4(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_33(x) + F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_15(x) + F_16(x) + F_47(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_26(x) + F_54(x))
Eq(F_54(x), F_52(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": [1, 0, 2, 3], "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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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": [1, 0, 2, 3], "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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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