0132_0231_1230_1302_3012
Counting sequence:
1, 1, 2, 6, 19, 55, 149, 391, 1015, 2632, 6840, 17821, 46522, 121601, 318084, 832390, 2178749, 5703431, 14931015, 39088967, 102335105, 267915416, 701410042, 1836313421, 4807528724, 12586271025, 32951282374, 86267573846, 225851436615, 591286733127, 1548008759545, 4052739541911, 10610209862187, 27777890040216, 72723460253564, 190392490715085, 498454011885774, 1304969544935761, 3416454622914440, 8944394323799862, 23416728348476785, 61305790721621431, 160500643816377707, 420196140727501111, 1100087778366114229, 2880067194370829320, 7540113804746360574, 19740274219868238301, 51680708854858339240, 135301852344706763297, 354224848179261933450, 927372692193079018726, 2427893228399975103227, 6356306993006846270231, 16641027750620563685469, 43566776258854844762855, 114059301025943970578399, 298611126818977066946216, 781774079430987230232640, 2046711111473984623722557, 5358359254990966640904290, 14028366653498915298957921, 36726740705505779255935372, 96151855463018422468812326, 251728825683549488150463909, 659034621587630041982539815, 1725375039079340637797113999, 4517090495650391871408758631, 11825896447871834976429116265, 30960598847965113057878542392, 81055900096023504197206460930, 212207101440105399533740788141, 555565404224292694404015848892, 1454489111232772683678306701521, 3807901929474025356630904196174, 9969216677189303386214405824950, 26099748102093884802012313213999, 68330027629092351019822533749671, 178890334785183168257455287964865, 468340976726457153752543330071927, 1226132595394188293000174702174995, 3210056809456107725247980776374136, 8404037832974134882743767626865412, 22002056689466296922983322104136941, 57602132235424755886206198685457014, 150804340016807970735635273952142385, 394810887814999156320699623170875024, 1033628323428189498226463595560384086, 2706074082469569338358691163510175065, 7084593923980518516849609894970035287, 18547707689471986212190138521399821235, 48558529144435440119720805669229315031, 127127879743834334146972278486288006557, 332825110087067562321196029789634583336, 871347450517368352816615810882615618054, 2281217241465037496128651402858212141245, 5972304273877744135569338397692020671824, 15635695580168194910579363790217849736001, 40934782466626840596168752972961528393490, 107168651819712326877926895128666735297222, 280571172992510140037611932413038677346275
Generating function in Maple syntax:
(x^6-4*x^5+8*x^4-15*x^3+14*x^2-6*x+1)/(x^2-3*x+1)/(x-1)^4
Generating function in latex syntax:
\frac{x^{6}-4 x^{5}+8 x^{4}-15 x^{3}+14 x^{2}-6 x +1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**6 - 4*x**5 + 8*x**4 - 15*x**3 + 14*x**2 - 6*x + 1)/((x - 1)**4*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^4*F(x)-x^6+4*x^5-8*x^4+15*x^3-14*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)-x^{6}+4 x^{5}-8 x^{4}+15 x^{3}-14 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/6*n^3-n^2+5/6*n-1/5*(3/2+1/2*5^(1/2))^(-n)*5^(1/2)+1/5*(3/2-1/2*5^(1/2))^(-n)*5^(1/2))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{n^{3}}{6}-n^{2}+\frac{5 n}{6}-\frac{\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}+\frac{\left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5} & \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) = 55
a(6) = 149
a(n+2) = -1/6*n^3+1/2*n^2-a(n)+3*a(n+1)+5/3*n-1, 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) = 19
a \! \left(5\right) = 55
a \! \left(6\right) = 149
a \! \left(n +2\right) = -\frac{n^{3}}{6}+\frac{n^{2}}{2}-a \! \left(n \right)+3 a \! \left(n +1\right)+\frac{5 n}{3}-1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19023/
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[26,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[22,x]
F[22,x] = F[14,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[10,x]+F[23,x]
F[26,x] = F[27,x]+F[37,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[16,x]+F[30,x]
F[30,x] = F[27,x]+F[31,x]
F[31,x] = F[18,x]+F[32,x]+F[36,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[14,x]
F[35,x] = F[31,x]
F[36,x] = F[12,x]*F[30,x]
F[37,x] = F[18,x]+F[38,x]+F[58,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[40,x]+F[42,x]
F[40,x] = F[41,x]+F[6,x]
F[41,x] = F[24,x]
F[42,x] = F[37,x]+F[43,x]
F[43,x] = 2*F[18,x]+F[44,x]+F[48,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[41,x]
F[47,x] = F[43,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[51,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[25,x]+F[49,x]
F[53,x] = 2*F[18,x]+F[48,x]+F[54,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[23,x]
F[57,x] = F[53,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[27,x]+F[50,x]
F[61,x] = F[31,x]+F[62,x]
F[62,x] = 2*F[18,x]+F[54,x]+F[63,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[50,x]+F[62,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_{26}\! \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_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{14}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right)+F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{37}\! \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_{16}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{18}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \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_{14}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)+F_{38}\! \left(x \right)+F_{58}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{6}\! \left(x \right)
F_{41}\! \left(x \right) = F_{24}\! \left(x \right)
F_{42}\! \left(x \right) = F_{37}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{48}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{41}\! \left(x \right)
F_{47}\! \left(x \right) = F_{43}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{25}\! \left(x \right)+F_{49}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \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_{23}\! \left(x \right)
F_{57}\! \left(x \right) = F_{53}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{27}\! \left(x \right)+F_{50}\! \left(x \right)
F_{61}\! \left(x \right) = F_{31}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{54}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{50}\! \left(x \right)+F_{62}\! \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_26(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_22(x))
Eq(F_22(x), F_14(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_10(x) + F_23(x))
Eq(F_26(x), F_27(x) + F_37(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_16(x) + F_30(x))
Eq(F_30(x), F_27(x) + F_31(x))
Eq(F_31(x), F_18(x) + F_32(x) + F_36(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_14(x))
Eq(F_35(x), F_31(x))
Eq(F_36(x), F_12(x)*F_30(x))
Eq(F_37(x), F_18(x) + F_38(x) + F_58(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_42(x))
Eq(F_40(x), F_41(x) + F_6(x))
Eq(F_41(x), F_24(x))
Eq(F_42(x), F_37(x) + F_43(x))
Eq(F_43(x), 2*F_18(x) + F_44(x) + F_48(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_41(x))
Eq(F_47(x), F_43(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_25(x) + F_49(x))
Eq(F_53(x), 2*F_18(x) + F_48(x) + F_54(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_23(x))
Eq(F_57(x), F_53(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_27(x) + F_50(x))
Eq(F_61(x), F_31(x) + F_62(x))
Eq(F_62(x), 2*F_18(x) + F_54(x) + F_63(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_50(x) + F_62(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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 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