0123_0231_1320_3012_3102
Counting sequence:
1, 1, 2, 6, 19, 49, 107, 212, 397, 719, 1276, 2236, 3886, 6715, 11554, 19812, 33873, 57763, 98269, 166814, 282595, 477829, 806514, 1359046, 2286584, 3841629, 6445552, 10800882, 18077887, 30224389, 50479871, 84228056, 140409961, 233864555, 389203864, 647225152, 1075516018, 1785985903, 2963834734, 4915399344, 8147146845, 13496038999, 22344592849, 36975533186, 61156435999, 101103182209, 167067143022, 275949065002, 455602474652, 751916548209, 1240470300316, 2045703136926, 3372441005611, 5557728001945, 9156020438099, 15079183732892, 24826490897413, 40862396652743, 67236896302324, 110604023733124, 181893659569270, 299055153618019, 491559023040634, 807781856831676, 1327118769902889, 2179846196942539, 3579688427088517, 5877203654486822, 9647284572279067, 15832549747930429, 25978288332082986, 42617353613056846, 69900611490062336, 114629450181090885, 187946516294053384, 308103906176022282, 504994816793860183, 827571056994551533, 1355982602136872039, 2221442721504559904, 3638731114363035745, 5959368688962343139, 9758600447141737744, 15977664633015195016, 26156461220884413754, 42814017491538212119, 70070566490788718662, 114664563398331635592, 187615197083491160757, 306939807092198316799, 502095117980435813881, 821235085487755997066, 1343070477688060023511, 2196245997810806109625, 3590997184353724445094, 5870864325654378966322, 9597163362352810145828, 15686950683841744269273, 25638338894373816318004, 41898437422229377658934, 68464149008796272963395
Generating function in Maple syntax:
-(2*x^9+6*x^8+3*x^7-5*x^6-4*x^5-3*x^4+2*x^3+5*x^2-4*x+1)/(x^2+x-1)^2/(x-1)^3
Generating function in latex syntax:
-\frac{2 x^{9}+6 x^{8}+3 x^{7}-5 x^{6}-4 x^{5}-3 x^{4}+2 x^{3}+5 x^{2}-4 x +1}{\left(x^{2}+x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-2*x**9 - 6*x**8 - 3*x**7 + 5*x**6 + 4*x**5 + 3*x**4 - 2*x**3 - 5*x**2 + 4*x - 1)/((x - 1)**3*(x**2 + x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(x^2+x-1)^2*(x-1)^3*F(x)+2*x^9+6*x^8+3*x^7-5*x^6-4*x^5-3*x^4+2*x^3+5*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+2 x^{9}+6 x^{8}+3 x^{7}-5 x^{6}-4 x^{5}-3 x^{4}+2 x^{3}+5 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/50*((-10*n-16)*5^(1/2)+20*n+50)*(-1/2*5^(1/2)-1/2)^(-n)+1/50*((10*n+16)*5^(1/2)+20*n+50)*(1/2*5^(1/2)-1/2)^(-n)+3/2*n^2-29/2*n+18)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(\left(-10 n -16\right) \sqrt{5}+20 n +50\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{50}+\\\frac{\left(\left(10 n +16\right) \sqrt{5}+20 n +50\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{50}+\frac{3 n^{2}}{2}-\frac{29 n}{2}+18 & \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) = 49
a(6) = 107
a(7) = 212
a(8) = 397
a(9) = 719
a(n+4) = 3/2*n^2-a(n)-2*a(n+1)+a(n+2)+2*a(n+3)-41/2*n+41, 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) = 49
a \! \left(6\right) = 107
a \! \left(7\right) = 212
a \! \left(8\right) = 397
a \! \left(9\right) = 719
a \! \left(n +4\right) = \frac{3 n^{2}}{2}-a \! \left(n \right)-2 a \! \left(n +1\right)+a \! \left(n +2\right)+2 a \! \left(n +3\right)-\frac{41 n}{2}+41, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17552/
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[23,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[18,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[14,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[22,x]
F[21,x] = F[11,x]
F[22,x] = F[18,x]
F[23,x] = F[24,x]+F[31,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[10,x]+F[27,x]
F[27,x] = F[24,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[24,x]
F[31,x] = F[32,x]+F[33,x]+F[64,x]
F[32,x] = 0
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[45,x]
F[35,x] = F[36,x]+F[7,x]
F[36,x] = F[32,x]+F[37,x]+F[40,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[11,x]+F[39,x]
F[39,x] = F[37,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[11,x]+F[39,x]
F[43,x] = F[44,x]
F[44,x] = F[21,x]*F[4,x]
F[45,x] = F[31,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[32,x]+F[50,x]+F[59,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[57,x]
F[52,x] = F[11,x]+F[53,x]
F[53,x] = F[32,x]+F[37,x]+F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[11,x]+F[56,x]
F[56,x] = F[44,x]
F[57,x] = F[49,x]+F[58,x]
F[58,x] = F[47,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[24,x]+F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[21,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[67,x]
F[66,x] = F[24,x]+F[49,x]
F[67,x] = F[68,x]+F[69,x]
F[68,x] = F[62,x]
F[69,x] = F[70,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[39,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_{23}\! \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_{18}\! \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_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)
F_{21}\! \left(x \right) = F_{11}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{31}\! \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_{10}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{24}\! \left(x \right)+F_{28}\! \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_{24}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{64}\! \left(x \right)
F_{32}\! \left(x \right) = 0
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{45}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{7}\! \left(x \right)
F_{36}\! \left(x \right) = F_{32}\! \left(x \right)+F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{11}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{11}\! \left(x \right)+F_{39}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{45}\! \left(x \right) = F_{31}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \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_{49}\! \left(x \right) = F_{32}\! \left(x \right)+F_{50}\! \left(x \right)+F_{59}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{57}\! \left(x \right)
F_{52}\! \left(x \right) = F_{11}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{32}\! \left(x \right)+F_{37}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{11}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{44}\! \left(x \right)
F_{57}\! \left(x \right) = F_{49}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{47}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{24}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{21}\! \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_{67}\! \left(x \right)
F_{66}\! \left(x \right) = F_{24}\! \left(x \right)+F_{49}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{69}\! \left(x \right)
F_{68}\! \left(x \right) = F_{62}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{39}\! \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_23(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_18(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_22(x))
Eq(F_21(x), F_11(x))
Eq(F_22(x), F_18(x))
Eq(F_23(x), F_24(x) + F_31(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_10(x) + F_27(x))
Eq(F_27(x), F_24(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_24(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_64(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_45(x))
Eq(F_35(x), F_36(x) + F_7(x))
Eq(F_36(x), F_32(x) + F_37(x) + F_40(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_11(x) + F_39(x))
Eq(F_39(x), F_37(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_11(x) + F_39(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_21(x)*F_4(x))
Eq(F_45(x), F_31(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_32(x) + F_50(x) + F_59(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_57(x))
Eq(F_52(x), F_11(x) + F_53(x))
Eq(F_53(x), F_32(x) + F_37(x) + F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_11(x) + F_56(x))
Eq(F_56(x), F_44(x))
Eq(F_57(x), F_49(x) + F_58(x))
Eq(F_58(x), F_47(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_24(x) + F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_21(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_67(x))
Eq(F_66(x), F_24(x) + F_49(x))
Eq(F_67(x), F_68(x) + F_69(x))
Eq(F_68(x), F_62(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_39(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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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, 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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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