0123_0132_1302_2031_2130
Counting sequence:
1, 1, 2, 6, 19, 53, 139, 353, 881, 2177, 5347, 13081, 31913, 77697, 188867, 458529, 1112097, 2695041, 6526787, 15797801, 38220729, 92435905, 223485795, 540193969, 1305446641, 3154233025, 7620204195, 18407224377, 44459818825, 107377193729, 259314869635, 626208259649, 1512134042177, 3651281650433, 8816307955843, 21287118787657, 51396987982169, 124093979653953, 299610717093923, 723366953449425, 1746447703207953, 4216468518295617, 10179797056659683, 24576887265335897, 59335220854773225, 143250627509765761, 345843072944071491, 834949967537442145, 2015769396298022497, 4866541536691620481, 11748958022797530051, 28364668688519213673, 68478717612301023481, 165322948338051392705, 399126303138264072931, 963578932314300066545, 2326290923166305261873, 5616174289445792701889, 13558666523655654888739, 32733561379952630925433, 79025897369951973631625, 190785572292638691972609, 460597474300793585144579, 1111981385585354317397121, 2684561974853759130209409, 6481108794057386398356993, 15646786480497559568005379, 37774695590110560816531593, 91196205330834791765396121, 220167161592012365475978817, 531530639195323964974663587, 1283228661343589179939925521, 3097988404604360093883753553, 7479206355996024905765910337, 18056402887483840981532529507, 43592015672738569021064879769, 105240441316510703328130109737, 254072912472859424286260740481, 613386294596428449118522873027, 1480845558334114116959049051041, 3575077524601452271908106103969, 8631000834210609838518231516545, 20837079646369854304430509652035, 50305161033644683158351131850409, 121447403527047950043076535412281, 293199971714518042088391726793665, 707847354209638951907635037236835, 1708894694640905781279211897741617, 4125636772505670185217159025668465, 9960168297680685493215730334975169, 24045973483923919854653020467411875, 58052115497642282568530572813384889, 140150204943435999723731769181353417, 338352526312969311480029317350435073, 816855259426284681611860816230910083, 1972063048879358792559891774509628097, 4760981364612642502443926014644911809, 11494025792959924268872307102588942593, 27749032980243051983037666817401778563, 66992091812867150120645893932550462665, 161733216724819595995725961072818629593
Generating function in Maple syntax:
(x^3+x^2+x-1)*(x^4-5*x^2+4*x-1)/(2*x-1)/(x^2+2*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{\left(x^{3}+x^{2}+x -1\right) \left(x^{4}-5 x^{2}+4 x -1\right)}{\left(2 x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**3 + x**2 + x - 1)*(x**4 - 5*x**2 + 4*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^3+x^2+x-1)*(x^4-5*x^2+4*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)+\left(x^{3}+x^{2}+x -1\right) \left(x^{4}-5 x^{2}+4 x -1\right) = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/8*(-2*2^(1/2)+4)*(-1-2^(1/2))^(-n)+1/8*(2*2^(1/2)+4)*(2^(1/2)-1)^(-n)-n-3/8*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-2 \sqrt{2}+4\right) \left(-1-\sqrt{2}\right)^{-n}}{8}+\frac{\left(2 \sqrt{2}+4\right) \left(\sqrt{2}-1\right)^{-n}}{8}-n -\frac{3 \,2^{n}}{8} & \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) = 53
a(6) = 139
a(7) = 353
a(n+3) = -2*a(n)-3*a(n+1)+4*a(n+2)-2*n+2, n >= 8
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) = 53
a \! \left(6\right) = 139
a \! \left(7\right) = 353
a \! \left(n +3\right) = -2 a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right)-2 n +2, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16577/
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[20,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[19,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[14,x]
F[19,x] = F[4,x]*F[7,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[13,x]+F[22,x]+F[51,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[26,x]
F[24,x] = F[25,x]+F[7,x]
F[25,x] = F[14,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[22,x]
F[28,x] = 2*F[13,x]+F[29,x]+F[33,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[25,x]
F[32,x] = F[29,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[46,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[18,x]+F[4,x]
F[40,x] = F[36,x]+F[41,x]
F[41,x] = 2*F[13,x]+F[33,x]+F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[18,x]
F[45,x] = F[42,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[50,x]
F[49,x] = F[36,x]
F[50,x] = F[47,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[56,x]
F[53,x] = F[2,x]+F[54,x]
F[54,x] = F[13,x]+F[37,x]+F[55,x]
F[55,x] = F[2,x]*F[4,x]
F[56,x] = F[57,x]+F[65,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[62,x]
F[60,x] = F[2,x]+F[61,x]
F[61,x] = F[55,x]
F[62,x] = F[57,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[57,x]
F[65,x] = 2*F[13,x]+F[47,x]+F[64,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_{20}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right)+F_{22}\! \left(x \right)+F_{51}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{14}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{22}\! \left(x \right)
F_{28}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{29}\! \left(x \right)+F_{33}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)
F_{32}\! \left(x \right) = F_{29}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{46}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{39}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{40}\! \left(x \right) = F_{36}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{33}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{18}\! \left(x \right)
F_{45}\! \left(x \right) = F_{42}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \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_{36}\! \left(x \right)
F_{50}\! \left(x \right) = F_{47}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{56}\! \left(x \right)
F_{53}\! \left(x \right) = F_{2}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{13}\! \left(x \right)+F_{37}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{65}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{2}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{55}\! \left(x \right)
F_{62}\! \left(x \right) = F_{57}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{65}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{47}\! \left(x \right)+F_{64}\! \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_20(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_19(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_4(x)*F_7(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_13(x) + F_22(x) + F_51(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_26(x))
Eq(F_24(x), F_25(x) + F_7(x))
Eq(F_25(x), F_14(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_22(x))
Eq(F_28(x), 2*F_13(x) + F_29(x) + F_33(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_25(x))
Eq(F_32(x), F_29(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_46(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_40(x))
Eq(F_39(x), F_18(x) + F_4(x))
Eq(F_40(x), F_36(x) + F_41(x))
Eq(F_41(x), 2*F_13(x) + F_33(x) + F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_18(x))
Eq(F_45(x), F_42(x))
Eq(F_46(x), F_47(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_36(x))
Eq(F_50(x), F_47(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_56(x))
Eq(F_53(x), F_2(x) + F_54(x))
Eq(F_54(x), F_13(x) + F_37(x) + F_55(x))
Eq(F_55(x), F_2(x)*F_4(x))
Eq(F_56(x), F_57(x) + F_65(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_62(x))
Eq(F_60(x), F_2(x) + F_61(x))
Eq(F_61(x), F_55(x))
Eq(F_62(x), F_57(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_57(x))
Eq(F_65(x), 2*F_13(x) + F_47(x) + F_64(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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