0123_0231_1230_2301_3012
Counting sequence:
1, 1, 2, 6, 19, 56, 151, 377, 886, 1989, 4316, 9136, 18997, 39006, 79389, 160611, 323616, 650307, 1304506, 2613874, 5233751, 10474836, 20958547, 41927741, 83868154, 167751281, 335520136, 671060772, 1342145321, 2684318074, 5368667641, 10737371271, 21474783492, 42949613391, 85899279174, 171798617286, 343597300651, 687194675152, 1374389432591, 2748778956609, 5497558014526, 10995116141021, 21990232405492, 43980464946776, 87960930042589, 175921860248406, 351843720675221, 703687441545067, 1407374883302056, 2814749766834459, 5629499533918866, 11258999068108506, 22517998136509887, 45035996273336076, 90071992547013259, 180143985094393861, 360287970189182786, 720575940378789897, 1441151880758034976, 2882303761516557644, 5764607523033637201, 11529215046067832306, 23058430092136260337, 46116860184273156111, 92233720368546989324, 184467440737094699431, 368934881474190165406, 737869762948381145262, 1475739525896763155091, 2951479051793527227144, 5902958103587055425991, 11805916207174111880841, 23611832414348224850182, 47223664828696450851061, 94447329657392902917644, 188894659314785807118336, 377789318629571615590021, 755578637259143232606542, 1511157274518286466715661, 3022314549036572935012979, 6044629098073145871689776, 12089258196146291745128691, 24178516392292583492095082, 48357032784585166986119746, 96714065569170333974264359, 193428131138340667950652356, 386856262276681335903530691, 773712524553362671809393357, 1547425049106725343621228426, 3094850098213450687245012129, 6189700196426901374492697016, 12379400392853802748988188276, 24758800785707605497979296377, 49517601571415210995961642346, 99035203142830421991926468329, 198070406285660843983856258711, 396140812571321687967715982356, 792281625142643375935435577087, 1584563250285286751870874918646, 3169126500570573503741753758614, 6338253001141147007483511600251
Generating function in Maple syntax:
-(3*x^5-12*x^4+18*x^3-15*x^2+6*x-1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
-\frac{3 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-3*x**5 + 12*x**4 - 18*x**3 + 15*x**2 - 6*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)+3*x^5-12*x^4+18*x^3-15*x^2+6*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)+3 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
-4+5*2^n-15/4*n-23/24*n^2-1/4*n^3-1/24*n^4
Explicit closed form in latex syntax:
-4+5 \,2^{n}-\frac{15 n}{4}-\frac{23 n^{2}}{24}-\frac{n^{3}}{4}-\frac{n^{4}}{24}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(n+1) = 2*a(n)+1/24*(n-1)*(n+4)*(n^2-n+6), 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) = 19
a \! \left(5\right) = 56
a \! \left(n +1\right) = 2 a \! \left(n \right)+\frac{\left(n -1\right) \left(n +4\right) \left(n^{2}-n +6\right)}{24}, \quad n \geq 6
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/21539/
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[5,x]
F[3,x] = F[1,x]+F[37,x]+F[4,x]
F[4,x] = F[5,x]*F[6,x]
F[5,x] = x
F[6,x] = F[19,x]+F[7,x]
F[7,x] = F[1,x]+F[4,x]+F[8,x]
F[8,x] = F[5,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[5,x]
F[13,x] = F[11,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[5,x]
F[17,x] = F[11,x]+F[14,x]
F[18,x] = F[13,x]*F[5,x]
F[19,x] = F[20,x]
F[20,x] = F[10,x]*F[21,x]*F[5,x]
F[21,x] = F[22,x]+F[26,x]+F[36,x]
F[22,x] = F[10,x]+F[23,x]
F[23,x] = F[11,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[23,x]*F[5,x]
F[26,x] = F[27,x]
F[27,x] = F[10,x]*F[28,x]*F[5,x]
F[28,x] = F[17,x]+F[29,x]
F[29,x] = F[10,x]*F[30,x]
F[30,x] = F[10,x]+F[31,x]
F[31,x] = F[11,x]+F[32,x]
F[32,x] = F[15,x]+F[33,x]+F[35,x]
F[33,x] = F[34,x]*F[5,x]
F[34,x] = F[11,x]+F[32,x]
F[35,x] = F[31,x]*F[5,x]
F[36,x] = F[21,x]*F[5,x]
F[37,x] = F[38,x]*F[5,x]
F[38,x] = F[39,x]+F[41,x]
F[39,x] = F[1,x]+F[37,x]+F[40,x]
F[40,x] = F[5,x]*F[9,x]
F[41,x] = F[42,x]
F[42,x] = F[43,x]*F[5,x]
F[43,x] = F[42,x]+F[44,x]+F[45,x]
F[44,x] = F[10,x]^2
F[45,x] = F[46,x]
F[46,x] = F[10,x]*F[47,x]*F[5,x]
F[47,x] = F[17,x]+F[48,x]
F[48,x] = F[10,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_{5}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{37}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right) F_{6}\! \left(x \right)
F_{5}\! \left(x \right) = x
F_{6}\! \left(x \right) = F_{19}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \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_{10}\! \left(x \right) F_{5}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \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_{17}\! \left(x \right) F_{5}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right) F_{5}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right) F_{21}\! \left(x \right) F_{5}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)+F_{36}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{23}\! \left(x \right) F_{5}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right) F_{28}\! \left(x \right) F_{5}\! \left(x \right)
F_{28}\! \left(x \right) = F_{17}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{10}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{11}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{5}\! \left(x \right)
F_{34}\! \left(x \right) = F_{11}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right) F_{5}\! \left(x \right)
F_{36}\! \left(x \right) = F_{21}\! \left(x \right) F_{5}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{5}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{41}\! \left(x \right)
F_{39}\! \left(x \right) = F_{1}\! \left(x \right)+F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{5}\! \left(x \right) F_{9}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right) F_{5}\! \left(x \right)
F_{43}\! \left(x \right) = F_{42}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10} \left(x \right)^{2}
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \left(x \right) F_{47}\! \left(x \right) F_{5}\! \left(x \right)
F_{47}\! \left(x \right) = F_{17}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \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)*F_5(x))
Eq(F_3(x), F_1(x) + F_37(x) + F_4(x))
Eq(F_4(x), F_5(x)*F_6(x))
Eq(F_5(x), x)
Eq(F_6(x), F_19(x) + F_7(x))
Eq(F_7(x), F_1(x) + F_4(x) + F_8(x))
Eq(F_8(x), F_5(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_5(x))
Eq(F_13(x), F_11(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_17(x)*F_5(x))
Eq(F_17(x), F_11(x) + F_14(x))
Eq(F_18(x), F_13(x)*F_5(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_10(x)*F_21(x)*F_5(x))
Eq(F_21(x), F_22(x) + F_26(x) + F_36(x))
Eq(F_22(x), F_10(x) + F_23(x))
Eq(F_23(x), F_11(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_23(x)*F_5(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_10(x)*F_28(x)*F_5(x))
Eq(F_28(x), F_17(x) + F_29(x))
Eq(F_29(x), F_10(x)*F_30(x))
Eq(F_30(x), F_10(x) + F_31(x))
Eq(F_31(x), F_11(x) + F_32(x))
Eq(F_32(x), F_15(x) + F_33(x) + F_35(x))
Eq(F_33(x), F_34(x)*F_5(x))
Eq(F_34(x), F_11(x) + F_32(x))
Eq(F_35(x), F_31(x)*F_5(x))
Eq(F_36(x), F_21(x)*F_5(x))
Eq(F_37(x), F_38(x)*F_5(x))
Eq(F_38(x), F_39(x) + F_41(x))
Eq(F_39(x), F_1(x) + F_37(x) + F_40(x))
Eq(F_40(x), F_5(x)*F_9(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_43(x)*F_5(x))
Eq(F_43(x), F_42(x) + F_44(x) + F_45(x))
Eq(F_44(x), F_10(x)**2)
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_10(x)*F_47(x)*F_5(x))
Eq(F_47(x), F_17(x) + F_48(x))
Eq(F_48(x), F_10(x)*F_30(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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": "row_and_col_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:
{"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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 2], [0, 2], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 2], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 2], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 0, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": 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