0213_0231_3012_3120
Counting sequence:
1, 1, 2, 6, 20, 62, 173, 441, 1049, 2371, 5164, 10952, 22794, 46820, 95303, 192803, 388455, 780545, 1565662, 3137002, 6280976, 12570426, 25151057, 50314301, 100643045, 201303087, 402626048, 805275196, 1610577094, 3221184896, 6442404939, 12884849927, 25769745299, 51539541965, 103079141778, 206158348478, 412316769580, 824633620150, 1649267330357, 3298534760577, 6597069631601, 13194139385051, 26388278904212, 52776557955696, 105553116072770, 211106232322012, 422212464836623, 844424929883051, 1688849859994239, 3377699720236121, 6755399440740614, 13510798881771602, 27021597763856904, 54043195528052210, 108086391056468953, 216172782113330053, 432345564227081405, 864691128454614855, 1729382256909714152, 3458764513819946852, 6917529027640448126, 13835058055281488376, 27670116110563608467, 55340232221127890191, 110680464442256497195, 221360928884513756837, 442721857769028323898, 885443715538057508006, 1770887431076115928484, 3541774862152232824046, 7083549724304466672189, 14167099448608934427977, 28334198897217870001609, 56668397794435741213555, 113336795588871483704828, 226673591177742968757528, 453347182355485938935930, 906694364710971879368660, 1813388729421943760313047, 3626777458843887522283827, 7253554917687775046310551, 14507109835375550094452401, 29014219670751100190827822, 58028439341502200383673786, 116056878683004400769464320, 232113757366008801541147562, 464227514732017603084619873, 928455029464035206171674061, 1856910058928070412345895829, 3713820117856140824694456671, 7427640235712281649391699664, 14855280471424563298786311052, 29710560942849126597575663414, 59421121885698253195154502000, 118842243771396506390312317403, 237684487542793012780628090903, 475368975085586025561259785155, 950737950171172051122523325565, 1901475900342344102245050563042, 3802951800684688204490105199502, 7605903601369376408980214638876
Generating function in Maple syntax:
-(x^2-x+1)*(4*x^3-9*x^2+5*x-1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
-\frac{\left(x^{2}-x +1\right) \left(4 x^{3}-9 x^{2}+5 x -1\right)}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-x**2 + x - 1)*(4*x**3 - 9*x**2 + 5*x - 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^5*F(x)+(x^2-x+1)*(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 -1\right)^{5} F \! \left(x \right)+\left(x^{2}-x +1\right) \left(4 x^{3}-9 x^{2}+5 x -1\right) = 0
Explicit closed form in Maple syntax:
-5+6*2^n-55/12*n-23/24*n^2-1/24*n^4-5/12*n^3
Explicit closed form in latex syntax:
-5+6 \,2^{n}-\frac{55 n}{12}-\frac{23 n^{2}}{24}-\frac{n^{4}}{24}-\frac{5 n^{3}}{12}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 62
a(n+1) = 2*a(n)+1/24*(n-1)*(n^3+7*n^2-6*n+24), n >= 6
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) = 62
a \! \left(n +1\right) = 2 a \! \left(n \right)+\frac{\left(n -1\right) \left(n^{3}+7 n^{2}-6 n +24\right)}{24}, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15728/
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[17,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[6,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[4,x]
F[12,x] = F[13,x]+F[16,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[4,x]
F[16,x] = F[11,x]
F[17,x] = F[18,x]+F[27,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[13,x]+F[21,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[14,x]
F[26,x] = F[22,x]
F[27,x] = F[28,x]+F[29,x]+F[72,x]
F[28,x] = 0
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[45,x]
F[31,x] = F[32,x]+F[7,x]
F[32,x] = F[28,x]+F[33,x]+F[35,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[7,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[14,x]
F[38,x] = F[39,x]
F[39,x] = F[28,x]+F[40,x]+F[41,x]
F[40,x] = F[25,x]*F[4,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[14,x]
F[44,x] = F[39,x]
F[45,x] = F[27,x]+F[46,x]
F[46,x] = 2*F[28,x]+F[47,x]+F[51,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[50,x]
F[49,x] = F[32,x]
F[50,x] = F[46,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[22,x]
F[54,x] = F[55,x]
F[55,x] = 2*F[28,x]+F[56,x]+F[68,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[62,x]
F[58,x] = F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[14,x]+F[59,x]
F[62,x] = F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[67,x]
F[66,x] = F[59,x]
F[67,x] = F[63,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[71,x]
F[70,x] = F[22,x]
F[71,x] = F[55,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[17,x]+F[74,x]
F[74,x] = F[75,x]
F[75,x] = F[28,x]+F[76,x]+F[80,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[78,x]+F[79,x]
F[78,x] = F[14,x]+F[59,x]
F[79,x] = F[22,x]+F[63,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[21,x]+F[82,x]
F[82,x] = F[75,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_{17}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{16}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{27}\! \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_{13}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{14}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{72}\! \left(x \right)
F_{28}\! \left(x \right) = 0
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{45}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{7}\! \left(x \right)
F_{32}\! \left(x \right) = F_{28}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{7}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{14}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{28}\! \left(x \right)+F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{14}\! \left(x \right)
F_{44}\! \left(x \right) = F_{39}\! \left(x \right)
F_{45}\! \left(x \right) = F_{27}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{28}\! \left(x \right)+F_{47}\! \left(x \right)+F_{51}\! \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_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{32}\! \left(x \right)
F_{50}\! \left(x \right) = F_{46}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{22}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = 2 F_{28}\! \left(x \right)+F_{56}\! \left(x \right)+F_{68}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{62}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{14}\! \left(x \right)+F_{59}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \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_{59}\! \left(x \right)
F_{67}\! \left(x \right) = F_{63}\! \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_{22}\! \left(x \right)
F_{71}\! \left(x \right) = F_{55}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{17}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{28}\! \left(x \right)+F_{76}\! \left(x \right)+F_{80}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{79}\! \left(x \right)
F_{78}\! \left(x \right) = F_{14}\! \left(x \right)+F_{59}\! \left(x \right)
F_{79}\! \left(x \right) = F_{22}\! \left(x \right)+F_{63}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{21}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{75}\! \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_17(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_6(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_4(x))
Eq(F_12(x), F_13(x) + F_16(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_4(x))
Eq(F_16(x), F_11(x))
Eq(F_17(x), F_18(x) + F_27(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_13(x) + F_21(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_14(x))
Eq(F_26(x), F_22(x))
Eq(F_27(x), F_28(x) + F_29(x) + F_72(x))
Eq(F_28(x), 0)
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_45(x))
Eq(F_31(x), F_32(x) + F_7(x))
Eq(F_32(x), F_28(x) + F_33(x) + F_35(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_7(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_14(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_28(x) + F_40(x) + F_41(x))
Eq(F_40(x), F_25(x)*F_4(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_14(x))
Eq(F_44(x), F_39(x))
Eq(F_45(x), F_27(x) + F_46(x))
Eq(F_46(x), 2*F_28(x) + F_47(x) + F_51(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_50(x))
Eq(F_49(x), F_32(x))
Eq(F_50(x), F_46(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_22(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), 2*F_28(x) + F_56(x) + F_68(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_62(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_14(x) + F_59(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_64(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_59(x))
Eq(F_67(x), F_63(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_22(x))
Eq(F_71(x), F_55(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_17(x) + F_74(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_28(x) + F_76(x) + F_80(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_78(x) + F_79(x))
Eq(F_78(x), F_14(x) + F_59(x))
Eq(F_79(x), F_22(x) + F_63(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_21(x) + F_82(x))
Eq(F_82(x), F_75(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:
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