0213_0231_1032_1203_3021
Counting sequence:
1, 1, 2, 6, 19, 55, 149, 392, 1022, 2662, 6943, 18135, 47417, 124056, 324674, 849870, 2224819, 5824447, 15248357, 39920432, 104512718, 273617470, 716339407, 1875400431, 4909861529, 12854183760, 33652689314, 88103883702, 230658961267, 603872999527, 1580960036693, 4139007109880, 10836061292222, 28369176766006, 74271469004959, 194445230247975, 509064221738009, 1332747434965032, 3489178083156002, 9134786814501822, 23915182360348243, 62610760266541615, 163917098439275237, 429140535051282656, 1123504506714571214, 2941372985092429390, 7700614448562715279, 20160470360595714687, 52780796633224426937, 138181919539077564192, 361764961984008263618, 947112966412947224550, 2479573937254833407827, 6491608845351552996631, 16995252598799825579669, 44494148951047923739880, 116487194254343945637374, 304967433811983913169542, 798415107181607793868447, 2090277887732839468432887, 5472418556016910611427193, 14326977780317892365845560, 37508514784936766486106242, 98198566574492407092469806, 257087184938540454791299699, 673062988241128957281425695, 1762101779784846417052973669, 4613242351113410293877491472, 12077625273555384464579496782, 31619633469552743099860994782, 82781275135102844835003483343, 216724191935755791405149450895, 567391300672164529380444864857, 1485449710080737796736185139056, 3888957829570048860828110547554, 10181423778629408785748146498710, 26655313506318177496416328943539, 69784516740325123703500840326727, 182698236714657193614086192031317, 478310193403646457138757735761752, 1252232343496282177802187015248318, 3278386837085200076267803309977430, 8582928167759318051001222914678047, 22470397666192754076735865434050631, 58828264830818944179206373387467609, 154014396826264078460883254728345800, 403214925647973291203443390797563234, 1055630380117655795149446917664337182, 2763676214704994094244897362195441427, 7235398263997326487585245168921980047, 18942518577286985368510838144570491493, 49592157467863629617947269264789487040, 129833953826303903485330969649797962062, 339909704011048080838045639684604391406, 889895158206840339028805949404015204239, 2329775770609472936248372208527441213215, 6099432153621578469716310676178308427129, 15968520690255262472900559820007484059712, 41806129917144208948985368783844143743362, 109449869061177364374055546531524947161542, 286543477266387884173181270810730697732243
Generating function in Maple syntax:
(x^6+2*x^5-2*x^4+6*x^3-9*x^2+5*x-1)/(x^2-3*x+1)/(x-1)^3
Generating function in latex syntax:
\frac{x^{6}+2 x^{5}-2 x^{4}+6 x^{3}-9 x^{2}+5 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**6 + 2*x**5 - 2*x**4 + 6*x**3 - 9*x**2 + 5*x - 1)/((x - 1)**3*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(x^2-3*x+1)*(x-1)^3*F(x)+x^6+2*x^5-2*x^4+6*x^3-9*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{6}+2 x^{5}-2 x^{4}+6 x^{3}-9 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/10*(49*5^(1/2)-105)*(3/2-1/2*5^(1/2))^(-n)+1/10*(-49*5^(1/2)-105)*(3/2+1/2*5^(1/2))^(-n)+n^2-8*n+14)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(49 \sqrt{5}-105\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(-49 \sqrt{5}-105\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+n^{2}-8 n +\\14 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 55
a(6) = 149
a(n+2) = -a(n)+3*a(n+1)-(-1+n)*(n-5), 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) = 19
a \! \left(5\right) = 55
a \! \left(6\right) = 149
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-\left(-1+n \right) \left(n -5\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20305/
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[17,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[13,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = F[1,x]+F[10,x]
F[13,x] = x
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = F[9,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[43,x]
F[19,x] = 0
F[20,x] = F[13,x]*F[21,x]
F[21,x] = F[22,x]+F[30,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[24,x]
F[24,x] = F[13,x]*F[25,x]
F[25,x] = F[1,x]+F[23,x]
F[26,x] = F[27,x]
F[27,x] = F[13,x]*F[28,x]
F[28,x] = F[10,x]+F[29,x]
F[29,x] = F[27,x]
F[30,x] = F[31,x]+F[34,x]
F[31,x] = F[19,x]+F[20,x]+F[32,x]
F[32,x] = F[13,x]*F[33,x]
F[33,x] = F[2,x]+F[31,x]
F[34,x] = F[35,x]
F[35,x] = F[13,x]*F[36,x]
F[36,x] = F[37,x]+F[42,x]
F[37,x] = F[38,x]
F[38,x] = F[13,x]*F[39,x]
F[39,x] = F[40,x]+F[41,x]
F[40,x] = F[13,x]
F[41,x] = F[38,x]
F[42,x] = F[35,x]
F[43,x] = F[13,x]*F[44,x]
F[44,x] = F[17,x]+F[45,x]
F[45,x] = F[37,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[13,x]*F[48,x]
F[48,x] = F[45,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_{17}\! \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_{13}\! \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_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{43}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{13}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{30}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \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_{42}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{13}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{13}\! \left(x \right)
F_{41}\! \left(x \right) = F_{38}\! \left(x \right)
F_{42}\! \left(x \right) = F_{35}\! \left(x \right)
F_{43}\! \left(x \right) = F_{13}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{17}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{37}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{13}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{45}\! \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_17(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_13(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), F_1(x) + F_10(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), F_9(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_43(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_13(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_30(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_13(x)*F_25(x))
Eq(F_25(x), F_1(x) + F_23(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_13(x)*F_28(x))
Eq(F_28(x), F_10(x) + F_29(x))
Eq(F_29(x), F_27(x))
Eq(F_30(x), F_31(x) + F_34(x))
Eq(F_31(x), F_19(x) + F_20(x) + F_32(x))
Eq(F_32(x), F_13(x)*F_33(x))
Eq(F_33(x), F_2(x) + F_31(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_13(x)*F_36(x))
Eq(F_36(x), F_37(x) + F_42(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_13(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_41(x))
Eq(F_40(x), F_13(x))
Eq(F_41(x), F_38(x))
Eq(F_42(x), F_35(x))
Eq(F_43(x), F_13(x)*F_44(x))
Eq(F_44(x), F_17(x) + F_45(x))
Eq(F_45(x), F_37(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_13(x)*F_48(x))
Eq(F_48(x), F_45(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, 2, 1, 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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 2, 1, 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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 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