0132_1203_3102_3120
Counting sequence:
1, 1, 2, 6, 20, 61, 162, 382, 819, 1626, 3030, 5354, 9042, 14687, 23062, 35154, 52201, 75732, 107610, 150078, 205808, 277953, 370202, 486838, 632799, 813742, 1036110, 1307202, 1635246, 2029475, 2500206, 3058922, 3718357, 4492584, 5397106, 6448950, 7666764, 9070917, 10683602, 12528942, 14633099, 17024386, 19733382, 22793050, 26238858, 30108903, 34444038, 39288002, 44687553, 50692604, 57356362, 64735470, 72890152, 81884361, 91785930, 102666726, 114602807, 127674582, 141966974, 157569586, 174576870, 193088299, 213208542, 235047642, 258721197, 284350544, 312062946, 341991782, 374276740, 409064013, 446506498, 486763998, 530003427, 576399018, 626132534, 679393482, 736379330, 797295727, 862356726, 931785010, 1005812121, 1084678692, 1168634682, 1257939614, 1352862816, 1453683665, 1560691834, 1674187542, 1794481807, 1921896702, 2056765614, 2199433506, 2350257182, 2509605555, 2677859918, 2855414218, 3042675333, 3240063352, 3448011858, 3666968214, 3897393852
Generating function in Maple syntax:
(x^8-3*x^7-x^6+9*x^5-20*x^4+22*x^3-16*x^2+6*x-1)/(x-1)^7
Generating function in latex syntax:
\frac{x^{8}-3 x^{7}-x^{6}+9 x^{5}-20 x^{4}+22 x^{3}-16 x^{2}+6 x -1}{\left(x -1\right)^{7}}
Generating function in sympy syntax:
(x**8 - 3*x**7 - x**6 + 9*x**5 - 20*x**4 + 22*x**3 - 16*x**2 + 6*x - 1)/(x - 1)**7
Implicit equation for the generating function in Maple syntax:
-(x-1)^7*F(x)+x^8-3*x^7-x^6+9*x^5-20*x^4+22*x^3-16*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right)^{7} F \! \left(x \right)+x^{8}-3 x^{7}-x^{6}+9 x^{5}-20 x^{4}+22 x^{3}-16 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/240*(-1+n)*(n^5-6*n^4+49*n^3-76*n^2-132*n+720))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-1+n \right) \left(n^{5}-6 n^{4}+49 n^{3}-76 n^{2}-132 n +720\right)}{240} & \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) = 61
a(6) = 162
a(7) = 382
a(8) = 819
a(n) = 1/240*(-1+n)*(n^5-6*n^4+49*n^3-76*n^2-132*n+720), n >= 9
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) = 61
a \! \left(6\right) = 162
a \! \left(7\right) = 382
a \! \left(8\right) = 819
a \! \left(n \right) = \frac{\left(-1+n \right) \left(n^{5}-6 n^{4}+49 n^{3}-76 n^{2}-132 n +720\right)}{240}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15607/
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[29,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[20,x]+F[26,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[11,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[22,x]*F[4,x]
F[25,x] = F[18,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[14,x]+F[28,x]
F[28,x] = F[26,x]
F[29,x] = F[30,x]+F[35,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[1,x]+F[14,x]
F[34,x] = F[30,x]
F[35,x] = F[19,x]+F[36,x]+F[58,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[57,x]
F[38,x] = F[39,x]+F[7,x]
F[39,x] = F[19,x]+F[20,x]+F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[27,x]+F[42,x]
F[42,x] = F[43,x]+F[49,x]
F[43,x] = F[19,x]+F[44,x]+F[46,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[22,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[17,x]+F[48,x]
F[48,x] = F[43,x]
F[49,x] = 2*F[19,x]+F[50,x]+F[54,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[28,x]
F[53,x] = F[49,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[43,x]+F[56,x]
F[56,x] = F[54,x]
F[57,x] = F[35,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[63,x]
F[60,x] = F[30,x]+F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[60,x]
F[63,x] = F[64,x]+F[72,x]
F[64,x] = F[19,x]+F[65,x]+F[68,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[14,x]+F[43,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[71,x]
F[70,x] = F[30,x]
F[71,x] = F[64,x]
F[72,x] = 2*F[19,x]+F[73,x]+F[77,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[75,x]+F[76,x]
F[75,x] = F[61,x]
F[76,x] = F[72,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[64,x]+F[79,x]
F[79,x] = F[77,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_{29}\! \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_{20}\! \left(x \right)+F_{26}\! \left(x \right)
F_{19}\! \left(x \right) = 0
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_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{11}\! \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_{18}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \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_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)
F_{35}\! \left(x \right) = F_{19}\! \left(x \right)+F_{36}\! \left(x \right)+F_{58}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{57}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{7}\! \left(x \right)
F_{39}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{27}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{49}\! \left(x \right)
F_{43}\! \left(x \right) = F_{19}\! \left(x \right)+F_{44}\! \left(x \right)+F_{46}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{22}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{17}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{43}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \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_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{28}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{43}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = F_{35}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{63}\! \left(x \right)
F_{60}\! \left(x \right) = F_{30}\! \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_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{72}\! \left(x \right)
F_{64}\! \left(x \right) = F_{19}\! \left(x \right)+F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{14}\! \left(x \right)+F_{43}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{71}\! \left(x \right)
F_{70}\! \left(x \right) = F_{30}\! \left(x \right)
F_{71}\! \left(x \right) = F_{64}\! \left(x \right)
F_{72}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{73}\! \left(x \right)+F_{77}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{76}\! \left(x \right)
F_{75}\! \left(x \right) = F_{61}\! \left(x \right)
F_{76}\! \left(x \right) = F_{72}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{64}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{77}\! \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_29(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) + F_20(x) + F_26(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_11(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_18(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_14(x) + F_28(x))
Eq(F_28(x), F_26(x))
Eq(F_29(x), F_30(x) + F_35(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_1(x) + F_14(x))
Eq(F_34(x), F_30(x))
Eq(F_35(x), F_19(x) + F_36(x) + F_58(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_57(x))
Eq(F_38(x), F_39(x) + F_7(x))
Eq(F_39(x), F_19(x) + F_20(x) + F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_27(x) + F_42(x))
Eq(F_42(x), F_43(x) + F_49(x))
Eq(F_43(x), F_19(x) + F_44(x) + F_46(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_22(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_17(x) + F_48(x))
Eq(F_48(x), F_43(x))
Eq(F_49(x), 2*F_19(x) + F_50(x) + F_54(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_28(x))
Eq(F_53(x), F_49(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_43(x) + F_56(x))
Eq(F_56(x), F_54(x))
Eq(F_57(x), F_35(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_63(x))
Eq(F_60(x), F_30(x) + F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_60(x))
Eq(F_63(x), F_64(x) + F_72(x))
Eq(F_64(x), F_19(x) + F_65(x) + F_68(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_14(x) + F_43(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_71(x))
Eq(F_70(x), F_30(x))
Eq(F_71(x), F_64(x))
Eq(F_72(x), 2*F_19(x) + F_73(x) + F_77(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_75(x) + F_76(x))
Eq(F_75(x), F_61(x))
Eq(F_76(x), F_72(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_64(x) + F_79(x))
Eq(F_79(x), F_77(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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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