0132_1032_1203_2301_2310
Counting sequence:
1, 1, 2, 6, 19, 50, 120, 275, 615, 1356, 2962, 6425, 13857, 29738, 63540, 135231, 286795, 606296, 1278054, 2687093, 5636229, 11796630, 24641704, 51380411, 106954959, 222298340, 461373690, 956301585, 1979711785, 4093641026, 8455717212, 17448305015, 35970351507, 74088186288, 152471339470, 313532613101, 644245094925, 1322849927726, 2714419331664, 5566277616243, 11407433138839, 23364622090940, 47828755808994, 97856534872841, 200111116256049, 409018325533530, 835628837110660, 1706442046309295, 3483252836795355, 7107243161945096, 14495961300599862, 29554872554620005, 60235645016081557, 122723089845847238, 249949779319063800, 508906757892867371, 1035827914295215455, 2107684625609393556, 4287426845256713674, 8718968878589281793, 17726168133330273849, 36028797018963969650, 73210515542534784684, 148726874094283261671, 302065434206993909539, 613354240450842593120, 1245155224975394736030, 2527203938098208573405, 5128194852491255351325, 10403963657572187113566, 21103075220323727050912, 42796446251006159751395, 86773484122729730804007, 175908151486894284212588, 356538669456658213636530, 722522071879055717698041, 1463933609689590016248385, 2965646151242137194203786, 6006850166210188711824084, 12164816059872206070483743, 24631863574648069434641259, 49868190059103453456632760, 100945305937821536087968774, 204308463514872330525346901, 413452630308203177749515429, 836576667173323388896677110, 1692496147460480844588649800, 3423677921148629822767893915, 6924727094752595912716979695, 14004196694415864359796346436, 28317878398653073788317470362, 57254726816948837714084499185, 115747393673183055703068118857, 233970667424936871955934482338, 472893095007015265011465457660, 955689710328313572222123905111, 1931186461285193228842633793715, 3901987003827518626482039558416, 7883202170169301590557623062894, 15924860665367131856302334022093, 32166633980791321062978843841069
Generating function in Maple syntax:
(2*x^7-9*x^6+7*x^5-6*x^4+14*x^3-14*x^2+6*x-1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
\frac{2 x^{7}-9 x^{6}+7 x^{5}-6 x^{4}+14 x^{3}-14 x^{2}+6 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(2*x**7 - 9*x**6 + 7*x**5 - 6*x**4 + 14*x**3 - 14*x**2 + 6*x - 1)/((x - 1)**3*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^2*(x-1)^3*F(x)+2*x^7-9*x^6+7*x^5-6*x^4+14*x^3-14*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+2 x^{7}-9 x^{6}+7 x^{5}-6 x^{4}+14 x^{3}-14 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/8*(2*n+3)*2^n+1/2*n^2-7/2*n+3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(2 n +3\right) 2^{n}}{8}+\frac{n^{2}}{2}-\frac{7 n}{2}+3 & \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) = 50
a(6) = 120
a(7) = 275
a(n+2) = 1/2*n^2-4*a(n)+4*a(n+1)-11/2*n+10, 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) = 50
a \! \left(6\right) = 120
a \! \left(7\right) = 275
a \! \left(n +2\right) = \frac{n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{11 n}{2}+10, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19707/
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[12,x]*F[4,x]
F[4,x] = F[31,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[26,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]+F[23,x]+F[25,x]
F[22,x] = 0
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[10,x]+F[21,x]
F[25,x] = F[12,x]*F[17,x]
F[26,x] = F[22,x]+F[25,x]+F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[10,x]
F[30,x] = F[21,x]
F[31,x] = F[32,x]+F[48,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[1,x]+F[14,x]
F[36,x] = F[18,x]+F[37,x]
F[37,x] = F[22,x]+F[23,x]+F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[17,x]+F[40,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[18,x]+F[41,x]
F[44,x] = 2*F[22,x]+F[45,x]+F[47,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[21,x]+F[44,x]
F[47,x] = F[12,x]*F[40,x]
F[48,x] = F[22,x]+F[49,x]+F[61,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[56,x]
F[51,x] = F[10,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[18,x]+F[55,x]
F[55,x] = F[53,x]
F[56,x] = F[21,x]+F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[41,x]+F[60,x]
F[60,x] = F[58,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[17,x]+F[63,x]
F[63,x] = F[37,x]+F[64,x]
F[64,x] = F[22,x]+F[47,x]+F[65,x]+F[66,x]
F[65,x] = 0
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[30,x]+F[68,x]
F[68,x] = F[44,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{31}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{26}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \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_{9}\! \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_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right)
F_{30}\! \left(x \right) = F_{21}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{48}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{36}\! \left(x \right) = F_{18}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{17}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{18}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{21}\! \left(x \right)+F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{48}\! \left(x \right) = F_{22}\! \left(x \right)+F_{49}\! \left(x \right)+F_{61}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{56}\! \left(x \right)
F_{51}\! \left(x \right) = F_{10}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{18}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{21}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{41}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{17}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{37}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{22}\! \left(x \right)+F_{47}\! \left(x \right)+F_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = 0
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{30}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{44}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_31(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_26(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_25(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_10(x) + F_21(x))
Eq(F_25(x), F_12(x)*F_17(x))
Eq(F_26(x), F_22(x) + F_25(x) + F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_10(x))
Eq(F_30(x), F_21(x))
Eq(F_31(x), F_32(x) + F_48(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_1(x) + F_14(x))
Eq(F_36(x), F_18(x) + F_37(x))
Eq(F_37(x), F_22(x) + F_23(x) + F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_17(x) + F_40(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_18(x) + F_41(x))
Eq(F_44(x), 2*F_22(x) + F_45(x) + F_47(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_21(x) + F_44(x))
Eq(F_47(x), F_12(x)*F_40(x))
Eq(F_48(x), F_22(x) + F_49(x) + F_61(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_56(x))
Eq(F_51(x), F_10(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_18(x) + F_55(x))
Eq(F_55(x), F_53(x))
Eq(F_56(x), F_21(x) + F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_41(x) + F_60(x))
Eq(F_60(x), F_58(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_17(x) + F_63(x))
Eq(F_63(x), F_37(x) + F_64(x))
Eq(F_64(x), F_22(x) + F_47(x) + F_65(x) + F_66(x))
Eq(F_65(x), 0)
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_30(x) + F_68(x))
Eq(F_68(x), F_44(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, 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": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 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, 0, 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": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 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