0231_0312_1230_2031
Counting sequence:
1, 1, 2, 6, 20, 68, 232, 792, 2704, 9232, 31520, 107616, 367424, 1254464, 4283008, 14623104, 49926400, 170459392, 581984768, 1987020288, 6784111616, 23162405888, 79081400320, 270000789504, 921840357376, 3147359850496, 10745758687232, 36688315047936, 125261742817280, 427670341173248, 1460157879058432, 4985290833887232, 17020847577432064, 58112808641953792, 198409539412951040, 677412540367896576, 2312831082645684224, 7896499249846943744, 26960334834096406528, 92048340836691738624, 314272693678574141440, 1072994093040913088512, 3663430984806504071168, 12507735753144190107648, 42704081042963752288256, 145800852665566628937728, 497795248576339011174400, 1699579288974222786822144, 5802726658744213124939776, 19811748057028406926114816, 67641538910625201454579712, 230942659528443991966089216, 788487560292525564955197440, 2692064922113214275888611328, 9191284567867805973644050432, 31381008427244795342798979072, 107141464573243569423907815424, 365803841438484687010033303552, 1248932436607451609192317583360, 4264122063552837062749203726336, 14558623380996445032612179738624, 49706249396880106004950311501824, 169707750825527533954576886530048, 579418504508349923808406923116544, 1978258516382344627324473919406080, 6754197056512678661681081831391232, 23060271193286025392075379486752768, 78732690660118744244939354284228608, 268810220253902926195606658163408896, 917775499695374216292547924085178368, 3133481558273691012778978380013895680, 10698375233704015618530817671885225984, 36526537818268680448565313927513112576, 124709400805666690557199620366281998336, 425784527586129401331667853610101768192, 1453719308733184224212272173707843076096, 4963308179760478094185752987611168768000, 16945794101575543928318467603028988919808, 57856560046781219524902364436893618143232, 197534651983973790242972522541516494733312, 674425487842332721922085361292278742646784, 2302632647401383307202396400086081981120512, 7861679613920867784965414877759770439188480, 26841453160880704525456866710866917794512896, 91642453415681082531896637087948130299674624, 312886907340962921076672814930058685609672704, 1068262722532489519242897985544338481839341568, 3647277075448032234818246312317236556138020864, 12452582856727149900787189278180269260873400320, 42515777276012535133512264488086603931217559552, 145157943390595840732474679395985877203123437568, 495600219010358292662874188607770300950058631168, 1692084989260241489186547395639109449393987649536, 5777139519020249371420441205340897195675833335808, 19724388097560514507308670030085369883915358044160, 67343273352201559286393797709659685144309765505024, 229924317213685208130957850778468000809408345931776, 785010722150337713951043807694552632949013852717056, 2680194254173980439542259529221274530177238719004672, 9150755572395246330266950501495992854810927170584576, 31242633781233024441983282947541422358889231244328960
Generating function in Maple syntax:
-(3*x-1)/(2*x^2-4*x+1)
Generating function in latex syntax:
-\frac{3 x -1}{2 x^{2}-4 x +1}
Generating function in sympy syntax:
(1 - 3*x)/(2*x**2 - 4*x + 1)
Implicit equation for the generating function in Maple syntax:
(2*x^2-4*x+1)*F(x)+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{2}-4 x +1\right) F \! \left(x \right)+3 x -1 = 0
Explicit closed form in Maple syntax:
-1/4*(2+2^(1/2))*(2*(1-1/2*2^(1/2))^(-n)*2^(1/2)-(1+1/2*2^(1/2))^(-n)-3*(1-1/2*2^(1/2))^(-n))
Explicit closed form in latex syntax:
-\frac{\left(2+\sqrt{2}\right) \left(2 \left(1-\frac{\sqrt{2}}{2}\right)^{-n} \sqrt{2}-\left(1+\frac{\sqrt{2}}{2}\right)^{-n}-3 \left(1-\frac{\sqrt{2}}{2}\right)^{-n}\right)}{4}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(n+2) = -2*a(n)+4*a(n+1), n >= 2
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(n +2\right) = -2 a \! \left(n \right)+4 a \! \left(n +1\right), \quad n \geq 2
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21410/
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[14,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[14,x]*F[7,x]
F[7,x] = F[46,x]+F[8,x]
F[8,x] = F[4,x]+F[9,x]
F[9,x] = F[10,x]*F[2,x]
F[10,x] = F[11,x]+F[27,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[15,x]
F[14,x] = x
F[15,x] = F[16,x]+F[19,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[14,x]*F[16,x]
F[19,x] = F[12,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[26,x]
F[21,x] = 0
F[22,x] = F[14,x]*F[23,x]
F[23,x] = F[24,x]+F[25,x]
F[24,x] = F[17,x]
F[25,x] = F[20,x]
F[26,x] = F[14,x]*F[19,x]
F[27,x] = F[17,x]+F[28,x]
F[28,x] = F[21,x]+F[29,x]+F[34,x]
F[29,x] = F[14,x]*F[30,x]
F[30,x] = F[17,x]+F[31,x]
F[31,x] = F[21,x]+F[29,x]+F[32,x]
F[32,x] = F[14,x]*F[33,x]
F[33,x] = F[17,x]+F[31,x]
F[34,x] = F[14,x]*F[35,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[17,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[14,x]*F[36,x]
F[39,x] = F[28,x]+F[40,x]
F[40,x] = 2*F[21,x]+F[41,x]+F[45,x]
F[41,x] = F[14,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[37,x]
F[44,x] = F[40,x]
F[45,x] = F[14,x]*F[39,x]
F[46,x] = F[47,x]
F[47,x] = F[0,x]*F[27,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_{14}\! \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_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{46}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{2}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{14}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{24}\! \left(x \right) = F_{17}\! \left(x \right)
F_{25}\! \left(x \right) = F_{20}\! \left(x \right)
F_{26}\! \left(x \right) = F_{14}\! \left(x \right) F_{19}\! \left(x \right)
F_{27}\! \left(x \right) = F_{17}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{21}\! \left(x \right)+F_{29}\! \left(x \right)+F_{34}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{17}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{21}\! \left(x \right)+F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{17}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{17}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right) F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{28}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{41}\! \left(x \right)+F_{45}\! \left(x \right)
F_{41}\! \left(x \right) = F_{14}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{37}\! \left(x \right)
F_{44}\! \left(x \right) = F_{40}\! \left(x \right)
F_{45}\! \left(x \right) = F_{14}\! \left(x \right) F_{39}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{0}\! \left(x \right) F_{27}\! \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_14(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_14(x)*F_7(x))
Eq(F_7(x), F_46(x) + F_8(x))
Eq(F_8(x), F_4(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_2(x))
Eq(F_10(x), F_11(x) + F_27(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_15(x))
Eq(F_14(x), x)
Eq(F_15(x), F_16(x) + F_19(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_14(x)*F_16(x))
Eq(F_19(x), F_12(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_26(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_14(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_25(x))
Eq(F_24(x), F_17(x))
Eq(F_25(x), F_20(x))
Eq(F_26(x), F_14(x)*F_19(x))
Eq(F_27(x), F_17(x) + F_28(x))
Eq(F_28(x), F_21(x) + F_29(x) + F_34(x))
Eq(F_29(x), F_14(x)*F_30(x))
Eq(F_30(x), F_17(x) + F_31(x))
Eq(F_31(x), F_21(x) + F_29(x) + F_32(x))
Eq(F_32(x), F_14(x)*F_33(x))
Eq(F_33(x), F_17(x) + F_31(x))
Eq(F_34(x), F_14(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_17(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_14(x)*F_36(x))
Eq(F_39(x), F_28(x) + F_40(x))
Eq(F_40(x), 2*F_21(x) + F_41(x) + F_45(x))
Eq(F_41(x), F_14(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_37(x))
Eq(F_44(x), F_40(x))
Eq(F_45(x), F_14(x)*F_39(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_0(x)*F_27(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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "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, 0, 3, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "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, 0, 3, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "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, 0, 3, 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": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 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