0231_1203_3021
Counting sequence:
1, 1, 2, 6, 21, 73, 241, 757, 2288, 6724, 19365, 54959, 154303, 429733, 1189430, 3276306, 8990037, 24591349, 67093357, 182653717, 496322396, 1346450176, 3647459397, 9868036571, 26666447611, 71984395333, 194127819746, 523052441742, 1408115003253, 3787828574689, 10181759025385, 27349808973109, 73417824133448, 196960697932348, 528083069455845, 1415083580339591, 3789925491549367, 10145154104494309, 27144128272766222, 72592475136639882, 194050301515873941, 518503923246401101, 1384880395595052517, 3697447452306879061, 9867971401234973876, 26326680484311576568, 70212193014437946501, 187190036122117256627, 498898169894148525043, 1329245029255173252229, 3540518189676182975930, 9427612518017989264134, 25096546464356860416693, 66789404070845298355513, 177699567983778264473185, 472665624887894850794101, 1256938370459323870535072, 3341716334806525643477044, 8882270079101385106304037, 23603705317546982894065247, 62710620529431303109518511, 166574868535142415665028901, 442372346636829919740133862, 1174570542640532277281684226, 3118066031213644728215593941, 8275779424910164403543423653, 21961001106093886117984204381, 58266258588746100287229894037, 154563149846797707971178960908, 409940281742445320677068527344, 1087083592418705899395161341125, 2882271095542229111283705291659, 7640785596665188180087983549163, 20252292800615807311383631302853, 53671658218851259414206216578450, 142217170986060209546391905929086, 376787756706582326444557567052277, 998115315886433936899409524406929, 2643657939205929096507511966153177, 7001188529662677158546606201408437, 18538797985171748457197076513316856, 49083546403787951186411795843244076, 129937973824004145034267744067923557, 343940431882516294920097933835643191, 910287359666190281165686858463435431, 2408923703824863658613780030454010213, 6374085884082511076789527565174557886, 16864138288516848794945774206202905722, 44613139869437776969279167210967325973, 118008909643534156620939536090984134909, 312119663144253232251540822362944709845, 825434673714443998689817921167719044309, 2182732065691026630108673840211410379108, 5771320052508023091913065926235337787368, 15258355971586780500091779259180083861765, 40336572972359193011312033965492934243171, 106622710396047780967918070616978106331107, 281812775457328415551085093625893140707589, 744787920249973666145935077333580424225642, 1968186680873077690447356559515440157363702, 5200706904836520071092417468886449927714101
Generating function in Maple syntax:
-(4*x^6-25*x^5+51*x^4-56*x^3+32*x^2-9*x+1)/(2*x-1)/(x-1)^2/(x^2-3*x+1)^2
Generating function in latex syntax:
-\frac{4 x^{6}-25 x^{5}+51 x^{4}-56 x^{3}+32 x^{2}-9 x +1}{\left(2 x -1\right) \left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2}}
Generating function in sympy syntax:
(-4*x**6 + 25*x**5 - 51*x**4 + 56*x**3 - 32*x**2 + 9*x - 1)/((x - 1)**2*(2*x - 1)*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^2*(x^2-3*x+1)^2*F(x)+4*x^6-25*x^5+51*x^4-56*x^3+32*x^2-9*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)+4 x^{6}-25 x^{5}+51 x^{4}-56 x^{3}+32 x^{2}-9 x +1 = 0
Explicit closed form in Maple syntax:
1/50*((-5*n-19)*5^(1/2)+15*n+75)*(3/2-1/2*5^(1/2))^(-n)+1/50*((5*n+19)*5^(1/2)+15*n+75)*(3/2+1/2*5^(1/2))^(-n)+2*n-2^(n+1)
Explicit closed form in latex syntax:
\frac{\left(\left(-5 n -19\right) \sqrt{5}+15 n +75\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{50}+\frac{\left(\left(5 n +19\right) \sqrt{5}+15 n +75\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{50}+2 n -2^{n +1}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 73
a(6) = 241
a(n+5) = 2*a(n)-13*a(n+1)+28*a(n+2)-23*a(n+3)+8*a(n+4)-2*n-2, 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) = 73
a \! \left(6\right) = 241
a \! \left(n +5\right) = 2 a \! \left(n \right)-13 a \! \left(n +1\right)+28 a \! \left(n +2\right)-23 a \! \left(n +3\right)+8 a \! \left(n +4\right)-2 n -2, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14764/
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[20,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[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[13,x]*F[17,x]
F[17,x] = F[14,x]+F[6,x]
F[18,x] = F[13,x]*F[19,x]
F[19,x] = F[9,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[15,x]+F[22,x]+F[49,x]
F[22,x] = F[13,x]*F[23,x]
F[23,x] = F[24,x]+F[30,x]
F[24,x] = F[25,x]+F[6,x]
F[25,x] = F[26,x]
F[26,x] = F[13,x]*F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[10,x]+F[29,x]
F[29,x] = F[26,x]
F[30,x] = F[21,x]+F[31,x]
F[31,x] = 2*F[15,x]+F[32,x]+F[36,x]
F[32,x] = F[13,x]*F[33,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[25,x]
F[35,x] = F[31,x]
F[36,x] = F[13,x]*F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]+F[44,x]
F[39,x] = F[40,x]
F[40,x] = F[13,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[10,x]
F[43,x] = F[39,x]
F[44,x] = 2*F[15,x]+F[36,x]+F[45,x]
F[45,x] = F[13,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[29,x]
F[48,x] = F[44,x]
F[49,x] = F[13,x]*F[50,x]
F[50,x] = F[20,x]+F[51,x]
F[51,x] = F[39,x]+F[52,x]
F[52,x] = 2*F[15,x]+F[45,x]+F[53,x]
F[53,x] = F[13,x]*F[54,x]
F[54,x] = F[51,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_{20}\! \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_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{9}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{49}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{30}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{6}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \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_{26}\! \left(x \right)
F_{30}\! \left(x \right) = F_{21}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{25}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{13}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{44}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{13}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{10}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{13}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{29}\! \left(x \right)
F_{48}\! \left(x \right) = F_{44}\! \left(x \right)
F_{49}\! \left(x \right) = F_{13}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{20}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{39}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{13}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{51}\! \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_20(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) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_13(x)*F_17(x))
Eq(F_17(x), F_14(x) + F_6(x))
Eq(F_18(x), F_13(x)*F_19(x))
Eq(F_19(x), F_9(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_15(x) + F_22(x) + F_49(x))
Eq(F_22(x), F_13(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_30(x))
Eq(F_24(x), F_25(x) + F_6(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_13(x)*F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_10(x) + F_29(x))
Eq(F_29(x), F_26(x))
Eq(F_30(x), F_21(x) + F_31(x))
Eq(F_31(x), 2*F_15(x) + F_32(x) + F_36(x))
Eq(F_32(x), F_13(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_25(x))
Eq(F_35(x), F_31(x))
Eq(F_36(x), F_13(x)*F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x) + F_44(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_13(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_10(x))
Eq(F_43(x), F_39(x))
Eq(F_44(x), 2*F_15(x) + F_36(x) + F_45(x))
Eq(F_45(x), F_13(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_29(x))
Eq(F_48(x), F_44(x))
Eq(F_49(x), F_13(x)*F_50(x))
Eq(F_50(x), F_20(x) + F_51(x))
Eq(F_51(x), F_39(x) + F_52(x))
Eq(F_52(x), 2*F_15(x) + F_45(x) + F_53(x))
Eq(F_53(x), F_13(x)*F_54(x))
Eq(F_54(x), F_51(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, 3, 1], "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, 3, 1], "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": 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