0132_0231_2031_3021_3120
Counting sequence:
1, 1, 2, 6, 19, 57, 160, 424, 1073, 2619, 6214, 14418, 32863, 73837, 163964, 360588, 786589, 1704111, 3670210, 7864534, 16777451, 35651841, 75497752, 159383856, 335544649, 704643427, 1476395390, 3087008154, 6442451383, 13421773269, 27917287924, 57982059028, 120259084853, 249108103767, 515396076154, 1065151890078, 2199023256259, 4535485465321, 9345848836880, 19241453486904, 39582418600801, 81363860456331, 167125767422902, 343047627867106, 703687441777679, 1442559255643197, 2955487255463020, 6051711999280284, 12384898975270093, 25332747903960319, 51791395714762034, 105834591243208038, 216172782113785243, 441352763482310097, 900719925474100744, 1837468647967163968, 3746994889972254329, 7638104968020362931, 15564440312192435950, 31705341376688293674, 64563604257983432551, 131433051525180557221, 267477789068788500452, 544178950174431774756, 1106804644422573099109, 2250502776992565299367, 4574792530279968803050, 9297159013149614016814, 18889465931478580857203, 38369227673315867363769, 77919046967349146028544, 158199277176133114661448, 321120920835135874534033, 651686574636011039492827, 1322262615203500659837734, 2682304162269958481382258, 5440166188265831286180799, 11031448103983491219196941, 22365127662870639732067420, 45334718235548594051484844, 91878362290711817277672701, 186174576220652892904754511, 377184855719764302508330402, 764041117996445638414306806, 1547425049106725343623908939, 3133535724441118820838411937, 6344442701337573908858015480, 12843627907585820352078417744, 25996740824992985772881612713, 52612451669628661683212783619, 106462843378542703641324687454, 215401566835656167832447619258, 435754893828453856764491731223, 881413307971190755728176451957, 1782633656570947595854738887124, 3604881394399027360506249744948, 7288990951312319058606043435669, 14736438227653166792399174767351, 29789789105363390935172525331290, 60213403510840896571093402260414, 121694457621910022543683507721251
Generating function in Maple syntax:
-(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{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:
(-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)+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)+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
5+1/2*n^2+3/2*n+2^n*n-4*2^n
Explicit closed form in latex syntax:
5+\frac{n^{2}}{2}+\frac{3 n}{2}+2^{n} n -4 \,2^{n}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+2) = 1/2*n^2-4*a(n)+4*a(n+1)-1/2*n+2, n >= 5
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(n +2\right) = \frac{n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{n}{2}+2, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21743/
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[13,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[28,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[23,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[14,x]
F[13,x] = x
F[14,x] = F[15,x]+F[18,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[13,x]*F[15,x]
F[18,x] = F[19,x]
F[19,x] = F[15,x]^2*F[13,x]*F[20,x]
F[20,x] = F[15,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[13,x]*F[15,x]*F[20,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[13,x]*F[23,x]
F[27,x] = F[25,x]
F[28,x] = F[29,x]+F[65,x]
F[29,x] = F[16,x]+F[30,x]
F[30,x] = F[31,x]+F[32,x]+F[35,x]
F[31,x] = 0
F[32,x] = F[13,x]*F[33,x]
F[33,x] = F[27,x]+F[34,x]
F[34,x] = F[32,x]
F[35,x] = F[13,x]*F[36,x]
F[36,x] = F[37,x]+F[46,x]
F[37,x] = F[16,x]+F[38,x]
F[38,x] = F[31,x]+F[39,x]+F[45,x]
F[39,x] = F[13,x]*F[40,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[42,x]
F[42,x] = F[13,x]*F[43,x]
F[43,x] = F[1,x]+F[41,x]
F[44,x] = F[39,x]
F[45,x] = F[13,x]*F[37,x]
F[46,x] = F[47,x]+F[53,x]
F[47,x] = F[31,x]+F[48,x]+F[51,x]
F[48,x] = F[13,x]*F[49,x]
F[49,x] = F[16,x]+F[50,x]
F[50,x] = F[48,x]
F[51,x] = F[13,x]*F[52,x]
F[52,x] = F[16,x]+F[47,x]
F[53,x] = F[31,x]+F[54,x]+F[62,x]+F[64,x]
F[54,x] = F[13,x]*F[55,x]
F[55,x] = F[56,x]+F[61,x]
F[56,x] = F[31,x]+F[57,x]+F[59,x]
F[57,x] = F[13,x]*F[58,x]
F[58,x] = F[41,x]+F[56,x]
F[59,x] = F[13,x]*F[60,x]
F[60,x] = F[16,x]+F[56,x]
F[61,x] = F[54,x]
F[62,x] = F[13,x]*F[63,x]
F[63,x] = F[38,x]+F[53,x]
F[64,x] = F[13,x]*F[46,x]
F[65,x] = F[30,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{28}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{23}\! \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_{13}\! \left(x \right) F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15} \left(x \right)^{2} F_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{23}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{65}\! \left(x \right)
F_{29}\! \left(x \right) = F_{16}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{13}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{27}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{13}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{46}\! \left(x \right)
F_{37}\! \left(x \right) = F_{16}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{31}\! \left(x \right)+F_{39}\! \left(x \right)+F_{45}\! \left(x \right)
F_{39}\! \left(x \right) = F_{13}\! \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_{13}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{1}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{39}\! \left(x \right)
F_{45}\! \left(x \right) = F_{13}\! \left(x \right) F_{37}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{53}\! \left(x \right)
F_{47}\! \left(x \right) = F_{31}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{13}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{16}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{13}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{16}\! \left(x \right)+F_{47}\! \left(x \right)
F_{53}\! \left(x \right) = F_{31}\! \left(x \right)+F_{54}\! \left(x \right)+F_{62}\! \left(x \right)+F_{64}\! \left(x \right)
F_{54}\! \left(x \right) = F_{13}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{61}\! \left(x \right)
F_{56}\! \left(x \right) = F_{31}\! \left(x \right)+F_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = F_{13}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{41}\! \left(x \right)+F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{13}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{16}\! \left(x \right)+F_{56}\! \left(x \right)
F_{61}\! \left(x \right) = F_{54}\! \left(x \right)
F_{62}\! \left(x \right) = F_{13}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{38}\! \left(x \right)+F_{53}\! \left(x \right)
F_{64}\! \left(x \right) = F_{13}\! \left(x \right) F_{46}\! \left(x \right)
F_{65}\! \left(x \right) = F_{30}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_13(x)*F_7(x))
Eq(F_7(x), F_28(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_23(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_14(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x) + F_18(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_13(x)*F_15(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_13(x)*F_15(x)**2*F_20(x))
Eq(F_20(x), F_15(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_13(x)*F_15(x)*F_20(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_13(x)*F_23(x))
Eq(F_27(x), F_25(x))
Eq(F_28(x), F_29(x) + F_65(x))
Eq(F_29(x), F_16(x) + F_30(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_35(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_13(x)*F_33(x))
Eq(F_33(x), F_27(x) + F_34(x))
Eq(F_34(x), F_32(x))
Eq(F_35(x), F_13(x)*F_36(x))
Eq(F_36(x), F_37(x) + F_46(x))
Eq(F_37(x), F_16(x) + F_38(x))
Eq(F_38(x), F_31(x) + F_39(x) + F_45(x))
Eq(F_39(x), F_13(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_13(x)*F_43(x))
Eq(F_43(x), F_1(x) + F_41(x))
Eq(F_44(x), F_39(x))
Eq(F_45(x), F_13(x)*F_37(x))
Eq(F_46(x), F_47(x) + F_53(x))
Eq(F_47(x), F_31(x) + F_48(x) + F_51(x))
Eq(F_48(x), F_13(x)*F_49(x))
Eq(F_49(x), F_16(x) + F_50(x))
Eq(F_50(x), F_48(x))
Eq(F_51(x), F_13(x)*F_52(x))
Eq(F_52(x), F_16(x) + F_47(x))
Eq(F_53(x), F_31(x) + F_54(x) + F_62(x) + F_64(x))
Eq(F_54(x), F_13(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_61(x))
Eq(F_56(x), F_31(x) + F_57(x) + F_59(x))
Eq(F_57(x), F_13(x)*F_58(x))
Eq(F_58(x), F_41(x) + F_56(x))
Eq(F_59(x), F_13(x)*F_60(x))
Eq(F_60(x), F_16(x) + F_56(x))
Eq(F_61(x), F_54(x))
Eq(F_62(x), F_13(x)*F_63(x))
Eq(F_63(x), F_38(x) + F_53(x))
Eq(F_64(x), F_13(x)*F_46(x))
Eq(F_65(x), F_30(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": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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