0123_0231_1302_2301
Counting sequence:
1, 1, 2, 6, 20, 63, 181, 480, 1199, 2871, 6676, 15210, 34150, 75839, 166967, 364956, 792777, 1712679, 3681838, 7880038, 16797800, 35678175, 75531401, 159426360, 335597779, 704709207, 1476476120, 3087106434, 6442570138, 13421915775, 27917457835, 57982260404, 120259322189, 249108382023, 515396400786, 1065152267070, 2199023692156, 4535485967263, 9345849412637, 19241454144912, 39582419350199, 81363861306999, 167125768385500, 343047628953114, 703687442999438, 1442559257013951, 2955487256996959, 6051712000992588, 12384898977176977, 25332747906079079, 51791395717111094, 105834591245806998, 216172782116654928, 441352763485472607, 900719925477579505, 1837468647970983784, 3746994889976441435, 7638104968024945047, 15564440312197442336, 31705341376693755186, 64563604257989381698, 131433051525187028223, 267477789068795529299, 544178950174439399268, 1106804644422581358997, 2250502776992574236295, 4574792530279978460698, 9297159013149624440942, 18889465931478592095716, 38369227673315879466783, 77919046967349159048453, 158199277176133128652992, 321120920835135889554367, 651686574636011055601591, 1322262615203500677097124, 2682304162269958499857098, 5440166188265831305938614, 11031448103983491240308031, 22365127662870639754604935, 45334718235548594075524860, 91878362290711817303294297, 186174576220652892932039847, 377184855719764302537364798, 764041117996445638445178822, 1547425049106725343656710456, 3133535724441118820873238239, 6344442701337573908894965337, 12843627907585820352117593496, 25996740824992985772923120355, 52612451669628661683256732887, 106462843378542703641371191912, 215401566835656167832496796386, 435754893828453856764543702506, 881413307971190755728231342975, 1782633656570947595854796827643, 3604881394399027360506310869012, 7288990951312319058606107881693, 14736438227653166792399242678215, 29789789105363390935172596854434, 60213403510840896571093477547934, 121694457621910022543683586929996
Generating function in Maple syntax:
-(6*x^7-36*x^6+78*x^5-97*x^4+75*x^3-35*x^2+9*x-1)/(2*x-1)^2/(x-1)^6
Generating function in latex syntax:
-\frac{6 x^{7}-36 x^{6}+78 x^{5}-97 x^{4}+75 x^{3}-35 x^{2}+9 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{6}}
Generating function in sympy syntax:
(-6*x**7 + 36*x**6 - 78*x**5 + 97*x**4 - 75*x**3 + 35*x**2 - 9*x + 1)/((x - 1)**6*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^6*F(x)+6*x^7-36*x^6+78*x^5-97*x^4+75*x^3-35*x^2+9*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{6} F \! \left(x \right)+6 x^{7}-36 x^{6}+78 x^{5}-97 x^{4}+75 x^{3}-35 x^{2}+9 x -1 = 0
Explicit closed form in Maple syntax:
5+2^n*n-4*2^n+29/20*n+1/120*n^5-1/24*n^4+1/24*n^3+13/24*n^2
Explicit closed form in latex syntax:
5+2^{n} n -4 \,2^{n}+\frac{29 n}{20}+\frac{n^{5}}{120}-\frac{n^{4}}{24}+\frac{n^{3}}{24}+\frac{13 n^{2}}{24}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 63
a(6) = 181
a(7) = 480
a(n+2) = 1/120*n^5-1/8*n^4+3/8*n^3+5/8*n^2-4*a(n)+4*a(n+1)-53/60*n+2, 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) = 20
a \! \left(5\right) = 63
a \! \left(6\right) = 181
a \! \left(7\right) = 480
a \! \left(n +2\right) = \frac{n^{5}}{120}-\frac{n^{4}}{8}+\frac{3 n^{3}}{8}+\frac{5 n^{2}}{8}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{53 n}{60}+2, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15017/
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[21,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[10,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[10,x]+F[14,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[10,x]+F[20,x]
F[20,x] = F[18,x]
F[21,x] = F[2,x]+F[22,x]
F[22,x] = F[15,x]+F[23,x]+F[54,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[34,x]
F[25,x] = F[26,x]+F[6,x]
F[26,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[31,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[10,x]+F[31,x]
F[34,x] = F[22,x]+F[35,x]
F[35,x] = 2*F[15,x]+F[36,x]+F[40,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[26,x]
F[39,x] = F[35,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[48,x]
F[42,x] = F[43,x]
F[43,x] = F[15,x]+F[44,x]+F[46,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[10,x]+F[43,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[10,x]+F[43,x]
F[48,x] = F[49,x]
F[49,x] = 2*F[15,x]+F[50,x]+F[52,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[31,x]+F[49,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[43,x]+F[49,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[47,x]+F[56,x]
F[56,x] = F[43,x]+F[57,x]
F[57,x] = F[15,x]+F[50,x]+F[58,x]+F[60,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[43,x]+F[57,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[20,x]+F[62,x]
F[62,x] = F[60,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_{21}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{2}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{54}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{34}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{6}\! \left(x \right)
F_{26}\! \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_{31}\! \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_{10}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{22}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{26}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{48}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{46}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{10}\! \left(x \right)+F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{10}\! \left(x \right)+F_{43}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{31}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{43}\! \left(x \right)+F_{49}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{47}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{43}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{43}\! \left(x \right)+F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{20}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{60}\! \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_21(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_10(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_10(x) + F_14(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_10(x) + F_20(x))
Eq(F_20(x), F_18(x))
Eq(F_21(x), F_2(x) + F_22(x))
Eq(F_22(x), F_15(x) + F_23(x) + F_54(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_34(x))
Eq(F_25(x), F_26(x) + F_6(x))
Eq(F_26(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_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_10(x) + F_31(x))
Eq(F_34(x), F_22(x) + F_35(x))
Eq(F_35(x), 2*F_15(x) + F_36(x) + F_40(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_26(x))
Eq(F_39(x), F_35(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_48(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_15(x) + F_44(x) + F_46(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_10(x) + F_43(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_10(x) + F_43(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), 2*F_15(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_31(x) + F_49(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_43(x) + F_49(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_47(x) + F_56(x))
Eq(F_56(x), F_43(x) + F_57(x))
Eq(F_57(x), F_15(x) + F_50(x) + F_58(x) + F_60(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_43(x) + F_57(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_20(x) + F_62(x))
Eq(F_62(x), F_60(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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