0132_0231_1230_2301
Counting sequence:
1, 1, 2, 6, 20, 66, 209, 633, 1843, 5194, 14262, 38372, 101635, 266035, 690341, 1780428, 4573136, 11717814, 29990501, 76746381, 196513111, 503749006, 1293238122, 3325633736, 8567271175, 22109841511, 57158273609, 148006024848, 383823124868, 996718908954, 2591435434217, 6744838268097, 17571407103355, 45813153606610, 119528751300126, 312038660886188, 815002388426251, 2129558300359579, 5566353888380141, 14553835324085076, 38061685698704696, 99559890955177086, 260466254562193517, 681516715824465141, 1783401096190353055, 4667241814467696406, 12215272102934030034, 31972135754242137488, 83687571584352137743, 219062044473050233039, 573438607664759191697, 1501127959206637678872, 3929681527901975960300, 10287364370594983895010, 26931256410578555192561, 70503990931740411096873, 184575676856659650513091, 483212528236708257704794, 1265040002344877592508806, 3311861866340898511438196, 8670450696826469990714131, 22699292930446035614015491, 59427018230916749117232437, 155580910906233811884613596, 407313949364961133453888160, 1066357277815793966344749126, 2791750302470606470954645109, 7308877931416818719690761053, 19134851007063535885597242343, 50095607906731872486913980574, 131151833846042694689640134202, 343359606747632377251059690072, 898926394035008742102417443095, 2353418352854770596176787943735, 6161326142785601194037402274521, 16130554876176535346456038600608, 42230327770694455214105723651156, 110560406363565889363139403537834, 289450845871948181737846950352121, 757792037711643363168469054628241, 1983925074817102997863778464993291, 5193982790967575434083640270917346, 13598022484478546703741707770502382, 35600082990523656150109334421948476, 93202223052534168221424810155097115, 244006579114205616882418490807512171, 638817499811987066721031665986275709, 1672445890611194640431549909572333668, 4378520111072392732882473590744156168, 11463114317612725216620131231439185934, 30010822585512194784904203199454687357, 78569352913727797471269896740971424197, 205697235079591818480088683067395853615, 538522350120895418021376573003325455910, 1409869810769565052349544553660373593506, 3691087072947815285801178716130732116768, 9663391389163108764930522515372801111071, 25299087055848457141086542957740644471199, 66233869699233328306579033101898725916961, 173402521879988391762008514644345242743336, 453973695609795812157364372702903921759676
Generating function in Maple syntax:
-(13*x^6-59*x^5+100*x^4-87*x^3+41*x^2-10*x+1)/(x^2-3*x+1)/(x-1)^2/(2*x-1)^3
Generating function in latex syntax:
-\frac{13 x^{6}-59 x^{5}+100 x^{4}-87 x^{3}+41 x^{2}-10 x +1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{2} \left(2 x -1\right)^{3}}
Generating function in sympy syntax:
(-13*x**6 + 59*x**5 - 100*x**4 + 87*x**3 - 41*x**2 + 10*x - 1)/((x - 1)**2*(2*x - 1)**3*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^2*(2*x-1)^3*F(x)+13*x^6-59*x^5+100*x^4-87*x^3+41*x^2-10*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{2} \left(2 x -1\right)^{3} F \! \left(x \right)+13 x^{6}-59 x^{5}+100 x^{4}-87 x^{3}+41 x^{2}-10 x +1 = 0
Explicit closed form in Maple syntax:
1/40*(4*5^(1/2)+20)*(3/2-1/2*5^(1/2))^(-n)+1/40*(-4*5^(1/2)+20)*(3/2+1/2*5^(1/2))^(-n)+2^(n+1)+1/40*(5*n^2-45*n)*2^n-n-2
Explicit closed form in latex syntax:
\frac{\left(4 \sqrt{5}+20\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{40}+\frac{\left(-4 \sqrt{5}+20\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{40}+2^{n +1}+\frac{\left(5 n^{2}-45 n \right) 2^{n}}{40}-n -2
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 66
a(6) = 209
a(n+5) = 8*a(n)-36*a(n+1)+50*a(n+2)-31*a(n+3)+9*a(n+4)-n, n >= 7
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) = 66
a \! \left(6\right) = 209
a \! \left(n +5\right) = 8 a \! \left(n \right)-36 a \! \left(n +1\right)+50 a \! \left(n +2\right)-31 a \! \left(n +3\right)+9 a \! \left(n +4\right)-n, \quad n \geq 7
Specification 1
Strategy pack name: point_and_row_and_col_placements
Tree: http://permpal.com/tree/21235/
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[7,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[7,x]*F[8,x]
F[7,x] = x
F[8,x] = F[48,x]+F[9,x]
F[9,x] = F[10,x]+F[26,x]+F[27,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[7,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[7,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[7,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]+F[23,x]+F[25,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[7,x]
F[24,x] = F[15,x]+F[21,x]
F[25,x] = F[17,x]*F[7,x]
F[26,x] = F[5,x]
F[27,x] = F[28,x]*F[7,x]
F[28,x] = F[29,x]+F[46,x]
F[29,x] = F[30,x]+F[36,x]
F[30,x] = F[20,x]+F[31,x]
F[31,x] = F[18,x]+F[32,x]
F[32,x] = F[22,x]+F[33,x]+F[35,x]
F[33,x] = F[34,x]*F[7,x]
F[34,x] = F[18,x]+F[32,x]
F[35,x] = F[31,x]*F[7,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[7,x]
F[38,x] = F[39,x]+F[45,x]
F[39,x] = F[40,x]+F[44,x]
F[40,x] = F[14,x]+F[41,x]+F[42,x]
F[41,x] = F[40,x]*F[7,x]
F[42,x] = F[43,x]
F[43,x] = F[14,x]*F[40,x]*F[7,x]
F[44,x] = F[14,x]*F[31,x]
F[45,x] = F[18,x]*F[40,x]
F[46,x] = F[47,x]
F[47,x] = F[13,x]*F[29,x]*F[7,x]
F[48,x] = F[49,x]
F[49,x] = F[7,x]*F[8,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_{7}\! \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_{7}\! \left(x \right) F_{8}\! \left(x \right)
F_{7}\! \left(x \right) = x
F_{8}\! \left(x \right) = F_{48}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{26}\! \left(x \right)+F_{27}\! \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_{7}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{7}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)
F_{25}\! \left(x \right) = F_{17}\! \left(x \right) F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{5}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{46}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{36}\! \left(x \right)
F_{30}\! \left(x \right) = F_{20}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{18}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{22}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{7}\! \left(x \right)
F_{34}\! \left(x \right) = F_{18}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right) F_{7}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{7}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{45}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{40}\! \left(x \right) = F_{14}\! \left(x \right)+F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{40}\! \left(x \right) F_{7}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{14}\! \left(x \right) F_{40}\! \left(x \right) F_{7}\! \left(x \right)
F_{44}\! \left(x \right) = F_{14}\! \left(x \right) F_{31}\! \left(x \right)
F_{45}\! \left(x \right) = F_{18}\! \left(x \right) F_{40}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{13}\! \left(x \right) F_{29}\! \left(x \right) F_{7}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{7}\! \left(x \right) F_{8}\! \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_7(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_7(x)*F_8(x))
Eq(F_7(x), x)
Eq(F_8(x), F_48(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_26(x) + F_27(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_7(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_7(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_7(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_25(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_7(x))
Eq(F_24(x), F_15(x) + F_21(x))
Eq(F_25(x), F_17(x)*F_7(x))
Eq(F_26(x), F_5(x))
Eq(F_27(x), F_28(x)*F_7(x))
Eq(F_28(x), F_29(x) + F_46(x))
Eq(F_29(x), F_30(x) + F_36(x))
Eq(F_30(x), F_20(x) + F_31(x))
Eq(F_31(x), F_18(x) + F_32(x))
Eq(F_32(x), F_22(x) + F_33(x) + F_35(x))
Eq(F_33(x), F_34(x)*F_7(x))
Eq(F_34(x), F_18(x) + F_32(x))
Eq(F_35(x), F_31(x)*F_7(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_7(x))
Eq(F_38(x), F_39(x) + F_45(x))
Eq(F_39(x), F_40(x) + F_44(x))
Eq(F_40(x), F_14(x) + F_41(x) + F_42(x))
Eq(F_41(x), F_40(x)*F_7(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_14(x)*F_40(x)*F_7(x))
Eq(F_44(x), F_14(x)*F_31(x))
Eq(F_45(x), F_18(x)*F_40(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_13(x)*F_29(x)*F_7(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_7(x)*F_8(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"}, {"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": "point_and_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, 3, 2], "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]]}], "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, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [2, 1]]}, 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