0231_1302_2103_2301
Counting sequence:
1, 1, 2, 6, 20, 62, 176, 473, 1238, 3210, 8315, 21585, 56182, 146551, 382855, 1001127, 2619291, 6855073, 17943713, 46973190, 122972185, 321938745, 842838312, 2206569146, 5776860565, 15124002242, 39595133856, 103661384748, 271389003238, 710505604920, 1860127788230, 4869877732855, 12749505379392, 33378638369916, 87386409690025, 228780590654407, 598955362221496, 1568085495951873, 4105301125568813, 10747817880681525, 28138152516394325, 73666639668410915, 192861766488738047, 504918659797692236, 1321894212904216235, 3460763978914821747, 9060397723840101086, 23720429192605319448, 62100889853975680063, 162582240369321527380, 425645831253988691470, 1114355253392644318050, 2917419928923944014152, 7637904533379187455106, 19996293671213618059820, 52350976480261666409637, 137056635769571380829626, 358818930828452475713598, 939400156715786045917863, 2459381539318905661617485, 6438744461240930938481290, 16856851844403887153340635, 44131811071970730521020707, 115538581371508304409165651, 302483933042554182705882655, 791913217756154243707849077, 2073255720225908548416989741, 5427853942921571401542401698, 14210306108538805656209451213, 37203064382694845567085139965, 97398887039545731045045106660, 254993596735942347568049265670, 667581903168281311659101721337, 1747752112768901587409254872246, 4575674435138423450568661809740, 11979271192646368764296729409192, 31362139142800682842321525205306, 82107146235755679762667844926748, 214959299564466356445682008224738, 562770752457643389574378178324195, 1473352957808463812277452525248580, 3857288120967748047257979395843280, 10098511405094780329496485660620917, 26438246094316592941231477584273891, 69216226877854998494197947090366700, 181210434539248402541362363684900357, 474415076739890209129889143962313321, 1242034795680422224848305068199919873, 3251689310301376465415026060635224313, 8513033135223707171396773113703425175, 22287410095369745048775293280472613675, 58349197150885527974929106727711864840, 152760181357286838876012026902660312447, 399931346920974988653106973980266282711, 1047033859405638127083308895038135620410, 2741170231295939392596819711134137533572, 7176476834482180050707150238364273801411, 18788260272150600759524631003958680553448, 49188303981969622227866742773511764398938, 128776651673758265924075597316576609036030, 337141651039305175544360049176218058949820
Generating function in Maple syntax:
(x^8-2*x^7-2*x^6+13*x^5-24*x^4+29*x^3-20*x^2+7*x-1)/(x^2-3*x+1)/(x-1)^5
Generating function in latex syntax:
\frac{x^{8}-2 x^{7}-2 x^{6}+13 x^{5}-24 x^{4}+29 x^{3}-20 x^{2}+7 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(x**8 - 2*x**7 - 2*x**6 + 13*x**5 - 24*x**4 + 29*x**3 - 20*x**2 + 7*x - 1)/((x - 1)**5*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(x^2-3*x+1)*(x-1)^5*F(x)+x^8-2*x^7-2*x^6+13*x^5-24*x^4+29*x^3-20*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}-3 x +1\right) \left(x -1\right)^{5} F \! \left(x \right)+x^{8}-2 x^{7}-2 x^{6}+13 x^{5}-24 x^{4}+29 x^{3}-20 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/120*(324*5^(1/2)-660)*(3/2-1/2*5^(1/2))^(-n)+1/120*(-324*5^(1/2)-660)*(3/2+1/2*5^(1/2))^(-n)+1/24*n^4-1/4*n^3+11/24*n^2-13/4*n+6)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(324 \sqrt{5}-660\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{\left(-324 \sqrt{5}-660\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{n^{4}}{24}-\frac{n^{3}}{4}\\+\frac{11 n^{2}}{24}-\frac{13 n}{4}+6 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 62
a(6) = 176
a(7) = 473
a(8) = 1238
a(n+2) = -a(n)+3*a(n+1)-1/24*(-1+n)*(n^3-n^2-14*n-72), n >= 9
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(5\right) = 62
a \! \left(6\right) = 176
a \! \left(7\right) = 473
a \! \left(8\right) = 1238
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-\frac{\left(-1+n \right) \left(n^{3}-n^{2}-14 n -72\right)}{24}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15920/
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[15,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[7,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]
F[14,x] = F[11,x]+F[7,x]
F[15,x] = F[16,x]+F[2,x]
F[16,x] = F[17,x]+F[18,x]+F[68,x]
F[17,x] = 0
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[20,x]+F[38,x]
F[20,x] = F[21,x]+F[7,x]
F[21,x] = F[17,x]+F[22,x]+F[24,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[20,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[33,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[1,x]+F[27,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[7,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[27,x]
F[37,x] = F[34,x]
F[38,x] = F[39,x]+F[49,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[44,x]
F[42,x] = F[27,x]+F[43,x]
F[43,x] = F[40,x]
F[44,x] = F[43,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[43,x]+F[45,x]
F[49,x] = 2*F[17,x]+F[50,x]+F[54,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[43,x]+F[53,x]
F[53,x] = 2*F[17,x]+F[50,x]+F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[63,x]
F[56,x] = F[57,x]+F[60,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[27,x]+F[57,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[43,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[67,x]
F[66,x] = F[57,x]
F[67,x] = F[64,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[74,x]
F[70,x] = F[27,x]+F[71,x]
F[71,x] = F[17,x]+F[68,x]+F[72,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[43,x]+F[7,x]
F[74,x] = F[43,x]+F[75,x]
F[75,x] = F[46,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_{15}\! \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_{7}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{11}\! \left(x \right)+F_{7}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{2}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{68}\! \left(x \right)
F_{17}\! \left(x \right) = 0
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{38}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{7}\! \left(x \right)
F_{21}\! \left(x \right) = F_{17}\! \left(x \right)+F_{22}\! \left(x \right)+F_{24}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{20}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{33}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{7}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{27}\! \left(x \right)
F_{37}\! \left(x \right) = F_{34}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{49}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{27}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{40}\! \left(x \right)
F_{44}\! \left(x \right) = F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{17}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{43}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{17}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \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_{63}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{60}\! \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_{27}\! \left(x \right)+F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{43}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{67}\! \left(x \right)
F_{66}\! \left(x \right) = F_{57}\! \left(x \right)
F_{67}\! \left(x \right) = F_{64}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{74}\! \left(x \right)
F_{70}\! \left(x \right) = F_{27}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{17}\! \left(x \right)+F_{68}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{43}\! \left(x \right)+F_{7}\! \left(x \right)
F_{74}\! \left(x \right) = F_{43}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{46}\! \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_15(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) + F_7(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_11(x) + F_7(x))
Eq(F_15(x), F_16(x) + F_2(x))
Eq(F_16(x), F_17(x) + F_18(x) + F_68(x))
Eq(F_17(x), 0)
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_20(x) + F_38(x))
Eq(F_20(x), F_21(x) + F_7(x))
Eq(F_21(x), F_17(x) + F_22(x) + F_24(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_20(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_33(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_1(x) + F_27(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_7(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_27(x))
Eq(F_37(x), F_34(x))
Eq(F_38(x), F_39(x) + F_49(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_44(x))
Eq(F_42(x), F_27(x) + F_43(x))
Eq(F_43(x), F_40(x))
Eq(F_44(x), F_43(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_43(x) + F_45(x))
Eq(F_49(x), 2*F_17(x) + F_50(x) + F_54(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_43(x) + F_53(x))
Eq(F_53(x), 2*F_17(x) + F_50(x) + F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_63(x))
Eq(F_56(x), F_57(x) + F_60(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_27(x) + F_57(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_43(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_67(x))
Eq(F_66(x), F_57(x))
Eq(F_67(x), F_64(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_74(x))
Eq(F_70(x), F_27(x) + F_71(x))
Eq(F_71(x), F_17(x) + F_68(x) + F_72(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_43(x) + F_7(x))
Eq(F_74(x), F_43(x) + F_75(x))
Eq(F_75(x), F_46(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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