0213_0231_3012_3102
Counting sequence:
1, 1, 2, 6, 20, 63, 186, 522, 1412, 3719, 9606, 24454, 61572, 153727, 381298, 940882, 2312196, 5663559, 13835902, 33728558, 82079428, 199460255, 484145642, 1174042970, 2844814468, 6888837703, 16672821494, 40335143958, 97544435972, 235826669055, 570003080418, 1377443442594, 3328111191044, 8040108275591, 19421212644078, 46908303367486, 113289358986564, 273590100555679, 660675718528090, 1595353854472234, 3852208061193348, 9301419244300551, 22458345084677734, 54224706483422630, 130920952191056260, 316092999144601727, 763159727038392914, 1842518006337654002, 4448406845946233860, 10739753910695187655, 25928759092266741086, 62598960945088933710, 151130058682165136324, 364865833708860262047, 880875236898767771850, 2126643329104160028666, 5134215938302616275076, 12395183292100449470663, 29924798695285629000150, 72245213028235935038518, 174416089442885954212356, 421079121296264753733631, 1016577790799929282220482, 2454241620425150959256386, 5925074866708286482896900, 14304419023957834489377543, 34533968254856176590306766, 83372466214134629927300702, 201279122044054320959527492, 485931153024101040875594399, 1173142313535971940769193786, 2832217550983475998530937034, 6837580957277786090064978052, 16507386549088772483128713607, 39852368222554779665258046278, 96212151328397229031516088262, 232276727547747032164032787076, 560765719760686882231066791039, 1353808393742711974369136626482, 3268382960593293186455280558738, 7890575221623663058251578773508, 19049535217229348724902200164935, 45989649282859819351943503221822, 111028841036503905116564254845550, 268047345862977464960622109386948, 647123561776678505788908666567583, 1562294527444773818040639828418474, 3771712732723104824874589095197338, 9105720225004740833798619562398852, 21983153646960101224489431307166535, 53072028447379972746812688351074998, 128127212398630105646185220358002774, 309326456958460301895323953764453124, 746780133743190945149114777281654271, 1802886739300122663618116807117252258, 4352553642053997215234474989095140066, 10507994082829238979785319980465495044, 25368541926554718946201621340342055559, 61245078173623164414981575441781457070, 147858698749170022861750797785168671614, 356962476622701160309655222134646204228
Generating function in Maple syntax:
-(x^4+2*x^3-8*x^2+5*x-1)/(2*x-1)/(x^2+2*x-1)/(x-1)^2
Generating function in latex syntax:
-\frac{x^{4}+2 x^{3}-8 x^{2}+5 x -1}{\left(2 x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**4 - 2*x**3 + 8*x**2 - 5*x + 1)/((x - 1)**2*(2*x - 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2+2*x-1)*(x-1)^2*F(x)+x^4+2*x^3-8*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{4}+2 x^{3}-8 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
1/2*n+2-5/8*(-1-2^(1/2))^(-n)*2^(1/2)+5/8*(2^(1/2)-1)^(-n)*2^(1/2)+(-1-2^(1/2))^(-n)-3*2^n+(2^(1/2)-1)^(-n)
Explicit closed form in latex syntax:
\frac{n}{2}+2-\frac{5 \left(-1-\sqrt{2}\right)^{-n} \sqrt{2}}{8}+\frac{5 \left(\sqrt{2}-1\right)^{-n} \sqrt{2}}{8}+\left(-1-\sqrt{2}\right)^{-n}-3 \,2^{n}+\left(\sqrt{2}-1\right)^{-n}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+3) = -2*a(n)-3*a(n+1)+4*a(n+2)+n+3, n >= 5
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(n +3\right) = -2 a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right)+n +3, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15727/
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[5,x]
F[4,x] = x
F[5,x] = F[17,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[6,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[4,x]
F[12,x] = F[13,x]+F[16,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[4,x]
F[16,x] = F[11,x]
F[17,x] = F[18,x]+F[24,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[13,x]+F[21,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[21,x]*F[4,x]
F[24,x] = F[25,x]+F[26,x]+F[58,x]
F[25,x] = 0
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[43,x]
F[28,x] = F[29,x]+F[7,x]
F[29,x] = F[25,x]+F[30,x]+F[32,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[7,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[14,x]
F[35,x] = F[36,x]
F[36,x] = F[25,x]+F[37,x]+F[39,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[14,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[14,x]
F[42,x] = F[36,x]
F[43,x] = F[24,x]+F[44,x]
F[44,x] = 2*F[25,x]+F[45,x]+F[47,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[24,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[50,x]
F[49,x] = F[22,x]
F[50,x] = F[51,x]
F[51,x] = 2*F[25,x]+F[52,x]+F[54,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[22,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[22,x]
F[57,x] = F[51,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[17,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[25,x]+F[62,x]+F[72,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[68,x]
F[64,x] = F[14,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[14,x]+F[65,x]
F[68,x] = F[22,x]+F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[22,x]+F[69,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[21,x]+F[74,x]
F[74,x] = F[61,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_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{17}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{16}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{24}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{58}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{43}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{7}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{25}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{14}\! \left(x \right)
F_{42}\! \left(x \right) = F_{36}\! \left(x \right)
F_{43}\! \left(x \right) = F_{24}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{24}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{22}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{22}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{22}\! \left(x \right)
F_{57}\! \left(x \right) = F_{51}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{17}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{25}\! \left(x \right)+F_{62}\! \left(x \right)+F_{72}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{68}\! \left(x \right)
F_{64}\! \left(x \right) = F_{14}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{14}\! \left(x \right)+F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = F_{22}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{22}\! \left(x \right)+F_{69}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{21}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{61}\! \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_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_17(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_6(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_4(x))
Eq(F_12(x), F_13(x) + F_16(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_4(x))
Eq(F_16(x), F_11(x))
Eq(F_17(x), F_18(x) + F_24(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_13(x) + F_21(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_21(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_58(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_43(x))
Eq(F_28(x), F_29(x) + F_7(x))
Eq(F_29(x), F_25(x) + F_30(x) + F_32(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_7(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_14(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_25(x) + F_37(x) + F_39(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_14(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_14(x))
Eq(F_42(x), F_36(x))
Eq(F_43(x), F_24(x) + F_44(x))
Eq(F_44(x), 2*F_25(x) + F_45(x) + F_47(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_24(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_50(x))
Eq(F_49(x), F_22(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), 2*F_25(x) + F_52(x) + F_54(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_22(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_22(x))
Eq(F_57(x), F_51(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_17(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_25(x) + F_62(x) + F_72(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_68(x))
Eq(F_64(x), F_14(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_14(x) + F_65(x))
Eq(F_68(x), F_22(x) + F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_22(x) + F_69(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_21(x) + F_74(x))
Eq(F_74(x), F_61(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:
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