0132_0213_1203_2130_3120
Counting sequence:
1, 1, 2, 6, 19, 55, 142, 331, 711, 1436, 2776, 5216, 9651, 17773, 32842, 61225, 115471, 220458, 425684, 829846, 1629995, 3219843, 6386374, 12703063, 25316311, 50518312, 100892784, 201606428, 402991843, 805713353, 1611098658, 3221802133, 6443131423, 12885700630, 25770736684, 51540692082, 103080470363, 206159877055, 412318521566, 824635620963, 1649269607527, 3298537343860, 6597072553096, 13194142679320, 26388282608403, 52776562109669, 105553120719226, 211106237506625, 422212470608175, 844424936293570, 1688849867099140, 3377699728094350, 6755399449414795, 13510798891328187, 27021597774366326, 54043195539589039, 108086391069112055, 216172782127162752, 432345564242191648, 864691128471095380, 1729382256927662659, 3458764513839466177, 6917529027661646418, 13835058055304479277, 27670116110588511295, 55340232221154830126, 110680464442285605468, 221360928884545170922, 442721857769062187707, 885443715538093972087, 1770887431076155150222, 3541774862152274967867, 7083549724304511909767, 14167099448608982938444, 28334198897217921971768, 56668397794435796838096, 113336795588871543186547, 226673591177743032307549, 453347182355486006773930, 906694364710971951723097, 1813388729421943837421391, 3626777458843887604392794, 7253554917687775133676340, 14507109835375550187340934, 29014219670751100289514987, 58028439341502200488445683, 116056878683004400880617510, 232113757366008801658989319, 464227514732017603209468439, 928455029464035206303858904, 1856910058928070412485757904, 3713820117856140824842348684, 7427640235712281649547986339, 14855280471424563298951369401, 29710560942849126597749883010, 59421121885698253195338285253, 118842243771396506390506079839, 237684487542793012780832261446, 475368975085586025561474806412, 950737950171172051122749654114, 1901475900342344102245288669723
Generating function in Maple syntax:
(x^7-2*x^6+16*x^5-30*x^4+33*x^3-21*x^2+7*x-1)/(2*x-1)/(x-1)^6
Generating function in latex syntax:
\frac{x^{7}-2 x^{6}+16 x^{5}-30 x^{4}+33 x^{3}-21 x^{2}+7 x -1}{\left(2 x -1\right) \left(x -1\right)^{6}}
Generating function in sympy syntax:
(x**7 - 2*x**6 + 16*x**5 - 30*x**4 + 33*x**3 - 21*x**2 + 7*x - 1)/((x - 1)**6*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^6*F(x)-x^7+2*x^6-16*x^5+30*x^4-33*x^3+21*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{6} F \! \left(x \right)-x^{7}+2 x^{6}-16 x^{5}+30 x^{4}-33 x^{3}+21 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-1+3*2^(-1+n)+1/40*n^5-1/6*n^4+13/24*n^3-4/3*n^2-1/15*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -1+3 \,2^{-1+n}+\frac{n^{5}}{40}-\frac{n^{4}}{6}+\frac{13 n^{3}}{24}-\frac{4 n^{2}}{3}-\frac{n}{15} & \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) = 55
a(6) = 142
a(7) = 331
a(n+1) = 2*a(n)-1/120*n*(-1+n)*(3*n^3-32*n^2+83*n-182), 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) = 55
a \! \left(6\right) = 142
a \! \left(7\right) = 331
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{n \left(-1+n \right) \left(3 n^{3}-32 n^{2}+83 n -182\right)}{120}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18427/
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[20,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[18,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[14,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[13,x]
F[20,x] = F[21,x]+F[26,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[25,x]
F[24,x] = F[1,x]+F[14,x]
F[25,x] = F[21,x]
F[26,x] = F[27,x]+F[28,x]+F[42,x]
F[27,x] = 0
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[41,x]
F[30,x] = F[31,x]+F[6,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[13,x]+F[34,x]
F[34,x] = F[35,x]+F[39,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[17,x]+F[38,x]
F[38,x] = F[35,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[34,x]
F[41,x] = F[26,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[44,x]+F[51,x]
F[44,x] = F[21,x]+F[45,x]
F[45,x] = F[27,x]+F[46,x]+F[50,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[10,x]+F[18,x]
F[49,x] = F[45,x]
F[50,x] = F[12,x]*F[44,x]
F[51,x] = F[52,x]+F[60,x]
F[52,x] = F[27,x]+F[53,x]+F[56,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[14,x]+F[35,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[21,x]
F[59,x] = F[52,x]
F[60,x] = 2*F[27,x]+F[61,x]+F[64,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[18,x]+F[39,x]
F[64,x] = F[12,x]*F[51,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_{20}\! \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_{18}\! \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_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{25}\! \left(x \right) = F_{21}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{42}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{41}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{6}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{39}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{17}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{35}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{41}\! \left(x \right) = F_{26}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{51}\! \left(x \right)
F_{44}\! \left(x \right) = F_{21}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{27}\! \left(x \right)+F_{46}\! \left(x \right)+F_{50}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{49}\! \left(x \right) = F_{45}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{60}\! \left(x \right)
F_{52}\! \left(x \right) = F_{27}\! \left(x \right)+F_{53}\! \left(x \right)+F_{56}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{14}\! \left(x \right)+F_{35}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{21}\! \left(x \right)
F_{59}\! \left(x \right) = F_{52}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{27}\! \left(x \right)+F_{61}\! \left(x \right)+F_{64}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{18}\! \left(x \right)+F_{39}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \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_20(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_18(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_14(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_13(x))
Eq(F_20(x), F_21(x) + F_26(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_25(x))
Eq(F_24(x), F_1(x) + F_14(x))
Eq(F_25(x), F_21(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_42(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_41(x))
Eq(F_30(x), F_31(x) + F_6(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_13(x) + F_34(x))
Eq(F_34(x), F_35(x) + F_39(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_17(x) + F_38(x))
Eq(F_38(x), F_35(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x)*F_34(x))
Eq(F_41(x), F_26(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_51(x))
Eq(F_44(x), F_21(x) + F_45(x))
Eq(F_45(x), F_27(x) + F_46(x) + F_50(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_10(x) + F_18(x))
Eq(F_49(x), F_45(x))
Eq(F_50(x), F_12(x)*F_44(x))
Eq(F_51(x), F_52(x) + F_60(x))
Eq(F_52(x), F_27(x) + F_53(x) + F_56(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_14(x) + F_35(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_21(x))
Eq(F_59(x), F_52(x))
Eq(F_60(x), 2*F_27(x) + F_61(x) + F_64(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_18(x) + F_39(x))
Eq(F_64(x), F_12(x)*F_51(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": [0, 2, 1, 3], "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, 1, 3, 0], "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": [0, 2, 1, 3], "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, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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