0213_0231_2301
Counting sequence:
1, 1, 2, 6, 21, 75, 261, 879, 2879, 9232, 29148, 90995, 281730, 866917, 2655218, 8103324, 24660429, 74878725, 226945401, 686801061, 2075857631, 6267709426, 18907559352, 56994786401, 171694156266, 516935557615, 1555647753926, 4679596639314, 14071875415389, 42302162252487, 127132875061653, 381990985656195, 1147523113207655, 3446626374127612, 10350497922174588, 31079288836427639, 93310624329268794, 280122334197991585, 840865594044806786, 2523902026557255216, 7575123084050485461, 22734314745986874801, 68226363165332346777, 204740399684765162121, 614381708494204874711, 1843565339215528157230, 5531796140609322661416, 16598268559391082976949, 49802345932715483631474, 149426778353841295827067, 448332016333328699224358, 1345131351978230711260470, 4035748283034671209380405, 12108172226299806334509507, 36326944581135018233350533, 108987190068412460055780279, 326978211269016797749965359, 980978200655366842132560520, 2943048661411241658444087996, 8829444595648774328550892571, 26489115561602214725186302866, 79469393397071039664789445501, 238413538552773952970222900402, 715254644029587043844395010916, 2145800658838690009349295705981, 6437498128389979790544065442045, 19312746114032516409267765851481, 57938897376792923834140118256909, 173818417505565424795647828251759, 521459769607487072683994246291722, 1564391134719499385697317873592888, 4693204364758526713825728910571945, 14079694150178037348222818760728730, 42239294657640274516550859668130727, 126718439538334492575312722310608678, 380156773104133599964553323819319994, 1140474127214368052850879559523800749, 3421432350859856905719750807300578655, 10264323152327823926771506262861717781, 30793037787011403104120442268412506251, 92379292251369598958439391575112760471, 277138345095086732258291541897842980276, 831416261417858008814806855126580445852, 2494251994310388318255424271877216450911, 7482764386969007600307712476432673778826, 22448315162963731610033174246686920680185, 67345003091023468940481636614393037999746, 202035160077410501000668100814720247237944, 606105875043119472743665533816318274876869, 1818318658757682155905504649257619109689577, 5454958682347129556255871949154158288695993, 16364883131635313887226171982452644735133745, 49094667942613633609544806708257005403353255, 147284052386370050215834696987097785035750022, 441852284286989904385358552216614040588124648, 1325557185686079820030678606411956309784814909, 3976672428405690017074469892983848608741903714, 11930019566434311595489068321867286277981664115, 35790064671607208822667665563468931713053421766, 107370209650517206953110557326827935731185655006, 322110669886334088119997568263351480903436808901
Generating function in Maple syntax:
-(11*x^6-50*x^5+88*x^4-81*x^3+40*x^2-10*x+1)/(3*x-1)/(2*x-1)/(x^2-3*x+1)/(x-1)^3
Generating function in latex syntax:
-\frac{11 x^{6}-50 x^{5}+88 x^{4}-81 x^{3}+40 x^{2}-10 x +1}{\left(3 x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-11*x**6 + 50*x**5 - 88*x**4 + 81*x**3 - 40*x**2 + 10*x - 1)/((x - 1)**3*(2*x - 1)*(3*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(3*x-1)*(2*x-1)*(x^2-3*x+1)*(x-1)^3*F(x)+11*x^6-50*x^5+88*x^4-81*x^3+40*x^2-10*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(3 x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+11 x^{6}-50 x^{5}+88 x^{4}-81 x^{3}+40 x^{2}-10 x +1 = 0
Explicit closed form in Maple syntax:
1/4*n^2+1/2*n+11/8+1/5*(3/2+1/2*5^(1/2))^(-n)*5^(1/2)-1/5*(3/2-1/2*5^(1/2))^(-n)*5^(1/2)-2^n+5/8*3^n
Explicit closed form in latex syntax:
\frac{n^{2}}{4}+\frac{n}{2}+\frac{11}{8}+\frac{\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}-\frac{\left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}-2^{n}+\frac{5 \,3^{n}}{8}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 75
a(6) = 261
a(n+4) = -6*a(n)+23*a(n+1)-22*a(n+2)+8*a(n+3)-1/2*n*(n+1), 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) = 21
a \! \left(5\right) = 75
a \! \left(6\right) = 261
a \! \left(n +4\right) = -6 a \! \left(n \right)+23 a \! \left(n +1\right)-22 a \! \left(n +2\right)+8 a \! \left(n +3\right)-\frac{n \left(n +1\right)}{2}, \quad n \geq 7
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/314/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[10,x]*F[3,x]
F[3,x] = F[0,x]+F[4,x]
F[4,x] = F[10,x]*F[5,x]
F[5,x] = F[20,x]+F[40,x]+F[6,x]
F[6,x] = F[11,x]+F[7,x]
F[7,x] = F[1,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[7,x]
F[10,x] = x
F[11,x] = F[12,x]+F[15,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[14,x]
F[14,x] = F[1,x]+F[12,x]
F[15,x] = F[16,x]+F[17,x]+F[19,x]
F[16,x] = 0
F[17,x] = F[10,x]*F[18,x]
F[18,x] = F[15,x]+F[8,x]
F[19,x] = F[10,x]*F[11,x]
F[20,x] = F[21,x]
F[21,x] = F[10,x]*F[22,x]
F[22,x] = 2*F[39,x]+F[23,x]+F[26,x]
F[23,x] = F[24,x]+F[6,x]
F[24,x] = F[25,x]
F[25,x] = F[10,x]*F[22,x]
F[26,x] = F[27,x]
F[27,x] = F[7,x]^2*F[10,x]*F[28,x]
F[28,x] = F[29,x]+F[31,x]
F[29,x] = F[1,x]+F[30,x]
F[30,x] = F[10,x]*F[28,x]
F[31,x] = F[32,x]
F[32,x] = F[10,x]*F[29,x]*F[33,x]
F[33,x] = F[14,x]+F[34,x]
F[34,x] = F[35,x]+F[8,x]
F[35,x] = F[16,x]+F[36,x]+F[38,x]
F[36,x] = F[10,x]*F[37,x]
F[37,x] = F[12,x]+F[35,x]
F[38,x] = F[10,x]*F[34,x]
F[39,x] = F[10,x]*F[22,x]
F[40,x] = F[10,x]*F[41,x]
F[41,x] = F[42,x]+F[5,x]
F[42,x] = F[43,x]
F[43,x] = F[10,x]*F[28,x]*F[44,x]*F[7,x]
F[44,x] = F[45,x]+F[7,x]
F[45,x] = F[46,x]+F[8,x]
F[46,x] = F[16,x]+F[47,x]+F[49,x]
F[47,x] = F[10,x]*F[48,x]
F[48,x] = F[46,x]+F[8,x]
F[49,x] = F[10,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_{10}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{0}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{10}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{20}\! \left(x \right)+F_{40}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{11}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{7}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{15}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{19}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{8}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = 2 F_{39}\! \left(x \right)+F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{6}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right) F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{7} \left(x \right)^{2} F_{10}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{31}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{10}\! \left(x \right) F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{10}\! \left(x \right) F_{29}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{14}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{8}\! \left(x \right)
F_{35}\! \left(x \right) = F_{16}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right)+F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right) F_{34}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right) F_{22}\! \left(x \right)
F_{40}\! \left(x \right) = F_{10}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{5}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \left(x \right) F_{28}\! \left(x \right) F_{44}\! \left(x \right) F_{7}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{7}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{8}\! \left(x \right)
F_{46}\! \left(x \right) = F_{16}\! \left(x \right)+F_{47}\! \left(x \right)+F_{49}\! \left(x \right)
F_{47}\! \left(x \right) = F_{10}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{46}\! \left(x \right)+F_{8}\! \left(x \right)
F_{49}\! \left(x \right) = F_{10}\! \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_10(x)*F_3(x))
Eq(F_3(x), F_0(x) + F_4(x))
Eq(F_4(x), F_10(x)*F_5(x))
Eq(F_5(x), F_20(x) + F_40(x) + F_6(x))
Eq(F_6(x), F_11(x) + F_7(x))
Eq(F_7(x), F_1(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_7(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_15(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_14(x))
Eq(F_14(x), F_1(x) + F_12(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_19(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_10(x)*F_18(x))
Eq(F_18(x), F_15(x) + F_8(x))
Eq(F_19(x), F_10(x)*F_11(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_10(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_26(x) + 2*F_39(x))
Eq(F_23(x), F_24(x) + F_6(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_10(x)*F_22(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_10(x)*F_28(x)*F_7(x)**2)
Eq(F_28(x), F_29(x) + F_31(x))
Eq(F_29(x), F_1(x) + F_30(x))
Eq(F_30(x), F_10(x)*F_28(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_10(x)*F_29(x)*F_33(x))
Eq(F_33(x), F_14(x) + F_34(x))
Eq(F_34(x), F_35(x) + F_8(x))
Eq(F_35(x), F_16(x) + F_36(x) + F_38(x))
Eq(F_36(x), F_10(x)*F_37(x))
Eq(F_37(x), F_12(x) + F_35(x))
Eq(F_38(x), F_10(x)*F_34(x))
Eq(F_39(x), F_10(x)*F_22(x))
Eq(F_40(x), F_10(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_5(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_10(x)*F_28(x)*F_44(x)*F_7(x))
Eq(F_44(x), F_45(x) + F_7(x))
Eq(F_45(x), F_46(x) + F_8(x))
Eq(F_46(x), F_16(x) + F_47(x) + F_49(x))
Eq(F_47(x), F_10(x)*F_48(x))
Eq(F_48(x), F_46(x) + F_8(x))
Eq(F_49(x), F_10(x)*F_45(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, 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": [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": [1, 0, 2], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, 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