0132_0231_1032_1203
Counting sequence:
1, 1, 2, 6, 20, 66, 213, 677, 2134, 6700, 21000, 65780, 206013, 645197, 2020718, 6328998, 19823284, 62090230, 194479961, 609155937, 1908020714, 5976378972, 18719461560, 58633880888, 183655502713, 575253474841, 1801833066090, 5643777070454, 17677675047492, 55370754381338, 173434596254429, 543239106687261, 1701556282660366, 5329685854310236, 16693865252975384, 52289223923089164, 163782497164327541, 513006397204146981, 1606860123208789302, 5033074576916173430, 15764806986589567684, 49379188709916381166, 154667562991718872353, 484456218637686620825, 1517434058163299983746, 4752970510625476688684, 14887453298774398908200, 46631104743383636396976, 146059899295699590507505, 457494933900249412816177, 1432984792907877198604562, 4488454985062100830091430, 14058926691083151863302964, 44035959002163314653255602, 137931275114345699602354149, 432034116793838748438222485, 1353235355209292834786275846, 4238660456211768288164588236, 13276509805845766736570184168, 41585240064794745006766511972, 130255030616944938502100851245, 407990262280213804645759169021, 1277924187089502741128614906430, 4002767661221128971687710198790, 12537636513640471444088153087380, 39270910193227640227381554945190, 123005989663736932050574521809129, 385284512599981507438466655180753, 1206804286971789104882857673942618, 3780002931406587280004841522467388, 11839883497013469502998989174329176, 37085378971039073074676290997670248, 116160377234497586699980771877147497, 363842398639043120676444206856954697, 1139642399578033178309933485653456378, 3569635654816749335693275560754791830, 11180962302610887363509031035359606500, 35021478408788323678964200164632621834, 109695741452488710267655446449022266861, 343593595688742774654073938656087191821, 1076218250910421664417527953020433552958, 3370975880010080050595251271764408430716, 10558711835630787353170799225747226919544, 33072439434825116323636273021853557288956, 103590879947982939024927882231508731756005, 324471692798616158759188336774911030305493, 1016323825808466108575911753459738703170182, 3183372053188741655577602382645316833485654, 9971091271979002630710642910695750179630884, 31231869694447270383124890112609359158769694, 97825770319855509402973794707334542680762545, 306413974965275839643310436852124123249379593, 959762685711907059356610862918416675140138610, 3006208881266988446864218030493751310341001100, 9416173364882463575525425106680003717681521992, 29493732584594626733429127919011382066428331104, 92381504467174498810337010386469176041259389921, 289361217443067868694318423880263627133274830433, 906349324391937949228303621651169122462047202082, 2838905313865522759327770257461941274620244748358, 8892138124007400825431529260276838239479857639764
Generating function in Maple syntax:
-(2*x-1)^3/(x-1)/(x^4+6*x^3-11*x^2+6*x-1)
Generating function in latex syntax:
-\frac{\left(2 x -1\right)^{3}}{\left(x -1\right) \left(x^{4}+6 x^{3}-11 x^{2}+6 x -1\right)}
Generating function in sympy syntax:
-(2*x - 1)**3/((x - 1)*(x**4 + 6*x**3 - 11*x**2 + 6*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^4+6*x^3-11*x^2+6*x-1)*F(x)+(2*x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{4}+6 x^{3}-11 x^{2}+6 x -1\right) F \! \left(x \right)+\left(2 x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
1/56*((9*I*2^(1/2)+13*I)*(-62+44*2^(1/2))^(1/2)-7*2^(1/2))*(-3/2+3/2*2^(1/2)-1/2*I*(-31+22*2^(1/2))^(1/2))^(-n)+1/56*((-9*I*2^(1/2)-13*I)*(-62+44*2^(1/2))^(1/2)-7*2^(1/2))*(-3/2+3/2*2^(1/2)+1/2*I*(-31+22*2^(1/2))^(1/2))^(-n)+1/56*((-13*2^(1/2)+18)*(31+22*2^(1/2))^(1/2)+7*2^(1/2))*(-3/2-3/2*2^(1/2)-1/2*(31+22*2^(1/2))^(1/2))^(-n)+1+1/56*((13*2^(1/2)-18)*(31+22*2^(1/2))^(1/2)+7*2^(1/2))*(-3/2-3/2*2^(1/2)+1/2*(31+22*2^(1/2))^(1/2))^(-n)
Explicit closed form in latex syntax:
\frac{\left(\left(9 \,\mathrm{I} \sqrt{2}+13 \,\mathrm{I}\right) \sqrt{-62+44 \sqrt{2}}-7 \sqrt{2}\right) \left(-\frac{3}{2}+\frac{3 \sqrt{2}}{2}-\frac{\mathrm{I} \sqrt{-31+22 \sqrt{2}}}{2}\right)^{-n}}{56}+\frac{\left(\left(-9 \,\mathrm{I} \sqrt{2}-13 \,\mathrm{I}\right) \sqrt{-62+44 \sqrt{2}}-7 \sqrt{2}\right) \left(-\frac{3}{2}+\frac{3 \sqrt{2}}{2}+\frac{\mathrm{I} \sqrt{-31+22 \sqrt{2}}}{2}\right)^{-n}}{56}+\frac{\left(\left(-13 \sqrt{2}+18\right) \sqrt{31+22 \sqrt{2}}+7 \sqrt{2}\right) \left(-\frac{3}{2}-\frac{3 \sqrt{2}}{2}-\frac{\sqrt{31+22 \sqrt{2}}}{2}\right)^{-n}}{56}+1+\frac{\left(\left(13 \sqrt{2}-18\right) \sqrt{31+22 \sqrt{2}}+7 \sqrt{2}\right) \left(-\frac{3}{2}-\frac{3 \sqrt{2}}{2}+\frac{\sqrt{31+22 \sqrt{2}}}{2}\right)^{-n}}{56}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = a(n)+6*a(n+1)-11*a(n+2)+6*a(n+3)-1, n >= 4
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(n +4\right) = a \! \left(n \right)+6 a \! \left(n +1\right)-11 a \! \left(n +2\right)+6 a \! \left(n +3\right)-1, \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15331/
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[12,x]*F[4,x]
F[4,x] = F[22,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[17,x]
F[21,x] = F[12,x]*F[13,x]
F[22,x] = F[2,x]+F[23,x]
F[23,x] = F[18,x]+F[24,x]+F[52,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[31,x]
F[26,x] = F[10,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[14,x]+F[30,x]
F[30,x] = F[28,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[18,x]+F[24,x]+F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[2,x]+F[32,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[51,x]
F[38,x] = F[18,x]+F[39,x]+F[49,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[12,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[14,x]
F[44,x] = F[45,x]+F[47,x]
F[45,x] = F[18,x]+F[39,x]+F[46,x]
F[46,x] = F[12,x]*F[2,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[38,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[2,x]+F[38,x]
F[51,x] = F[36,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[34,x]+F[54,x]
F[54,x] = F[38,x]+F[55,x]
F[55,x] = F[18,x]+F[56,x]+F[57,x]+F[59,x]
F[56,x] = 0
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[32,x]+F[55,x]
F[59,x] = F[12,x]*F[54,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{22}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{18}\! \left(x \right)+F_{24}\! \left(x \right)+F_{52}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \left(x \right)
F_{26}\! \left(x \right) = F_{10}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{14}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{18}\! \left(x \right)+F_{24}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{51}\! \left(x \right)
F_{38}\! \left(x \right) = F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{49}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{2}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{2}\! \left(x \right)+F_{38}\! \left(x \right)
F_{51}\! \left(x \right) = F_{36}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{34}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{38}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{18}\! \left(x \right)+F_{56}\! \left(x \right)+F_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{56}\! \left(x \right) = 0
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{32}\! \left(x \right)+F_{55}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_22(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_17(x))
Eq(F_21(x), F_12(x)*F_13(x))
Eq(F_22(x), F_2(x) + F_23(x))
Eq(F_23(x), F_18(x) + F_24(x) + F_52(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_31(x))
Eq(F_26(x), F_10(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_14(x) + F_30(x))
Eq(F_30(x), F_28(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_18(x) + F_24(x) + F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_2(x) + F_32(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_51(x))
Eq(F_38(x), F_18(x) + F_39(x) + F_49(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_12(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_14(x))
Eq(F_44(x), F_45(x) + F_47(x))
Eq(F_45(x), F_18(x) + F_39(x) + F_46(x))
Eq(F_46(x), F_12(x)*F_2(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_38(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_2(x) + F_38(x))
Eq(F_51(x), F_36(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_34(x) + F_54(x))
Eq(F_54(x), F_38(x) + F_55(x))
Eq(F_55(x), F_18(x) + F_56(x) + F_57(x) + F_59(x))
Eq(F_56(x), 0)
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_32(x) + F_55(x))
Eq(F_59(x), F_12(x)*F_54(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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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