0123_0132_0231_1230_1302
Counting sequence:
1, 1, 2, 6, 19, 57, 164, 461, 1280, 3530, 9695, 26551, 72552, 197881, 538814, 1464942, 3977419, 10785141, 29210060, 79023365, 213562940, 576595826, 1555312727, 4191676531, 11287631184, 30372625777, 81666558074, 219434575446, 589223437315, 1581187016625, 4240605183860, 11366479968701, 30450121452152, 81531893143706, 218198297516879, 583673209264687, 1560599220312504, 4170847911921001, 11142337006159670, 29754617118437310, 79426483894080955, 211941289186712301, 565341777989847452, 1507500773131297781, 4018466332125657524, 10708398867062041922, 28526926409787957959, 75972468140667933931, 202270545206586660000, 538379281283838392545, 1432607572864796740850, 3811124146165410153126, 10136066717976140464627, 26951300855942273155881, 71645208542043758002244, 190412217998588975933357, 505947752446730016046064, 1344072067092221635866986, 3569835225088789736293055, 9479492909200091543564455, 25167254629330461961318920, 66804045058222281570595801, 177291591656810367374160302, 470429089167199787192757006, 1248024042498287909503034539, 3310369779033169720572246117, 8779237150064239674682478444, 23279070496843098791625613541, 61717008962052686742176860460, 163597331428394302072702038290, 433592075818780611347337967415, 1149004792475819366945740932451, 3044382900456642602547763322352, 8065199808990131944894755443281, 21363423627953858631670243741034, 56580636479095736644519991571702, 149832974920566123985568037616099, 396726190212076660668815025410721, 1050314812392853161407091444376340, 2780317995068576708354471620864925, 7358969200441969314676145951897960, 19475479941042514403931421522720826, 51535811599412031050870661946260527, 136358087452587767041680739018156831, 360748507567807377799419535884502104, 954291473083808501239321636262129737, 2524127968373084422841528695005935654, 6675694564270869523171470647441042526, 17653760064456332117408518521269239195, 46680396516913125985374784539537451485, 123421057809711235336942298692903397180, 326288850994690149363810802702682809301, 862530089098339731271339719310114956452, 2279849123989801030662398493749061767906, 6025575812015498800835576567606299544999, 15924006191800529705991303487556124752347, 42079267873473157879459529924851709169984, 111185144879136312285203902097881618246081, 293757384005400816472280827771651357575650, 776059311410943881267207919614764475014470, 2050056245807599022239922294862859917061075
Generating function in Maple syntax:
(x^6-3*x^5+9*x^4-20*x^3+18*x^2-7*x+1)/(-1+x)^2/(x^2-3*x+1)^2
Generating function in latex syntax:
\frac{x^{6}-3 x^{5}+9 x^{4}-20 x^{3}+18 x^{2}-7 x +1}{\left(-1+x \right)^{2} \left(x^{2}-3 x +1\right)^{2}}
Generating function in sympy syntax:
(x**6 - 3*x**5 + 9*x**4 - 20*x**3 + 18*x**2 - 7*x + 1)/((x - 1)**2*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(-1+x)^2*(x^2-3*x+1)^2*F(x)-x^6+3*x^5-9*x^4+20*x^3-18*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(-1+x \right)^{2} \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)-x^{6}+3 x^{5}-9 x^{4}+20 x^{3}-18 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/50*((-15*n+19)*5^(1/2)+35*n-25)*(3/2-1/2*5^(1/2))^(-n)+1/50*((15*n-19)*5^(1/2)+35*n-25)*(3/2+1/2*5^(1/2))^(-n)-n+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-15 n +19\right) \sqrt{5}+35 n -25\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{50}+\\\frac{\left(\left(15 n -19\right) \sqrt{5}+35 n -25\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{50}-n +1 & \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) = 57
a(6) = 164
a(n+4) = -a(n)+6*a(n+1)-11*a(n+2)+6*a(n+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) = 57
a \! \left(6\right) = 164
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)-n -1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16112/
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[10,x]*F[4,x]
F[4,x] = F[20,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[11,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = x
F[11,x] = F[12,x]+F[15,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[14,x]
F[14,x] = F[1,x]+F[12,x]
F[15,x] = F[16,x]+F[17,x]+F[19,x]
F[16,x] = 0
F[17,x] = F[10,x]*F[18,x]
F[18,x] = F[10,x]+F[15,x]
F[19,x] = F[10,x]*F[12,x]
F[20,x] = F[21,x]+F[31,x]
F[21,x] = F[22,x]
F[22,x] = F[10,x]*F[23,x]
F[23,x] = F[14,x]+F[24,x]
F[24,x] = F[21,x]+F[25,x]
F[25,x] = F[16,x]+F[26,x]+F[30,x]
F[26,x] = F[10,x]*F[27,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[12,x]
F[29,x] = F[25,x]
F[30,x] = F[10,x]*F[24,x]
F[31,x] = F[16,x]+F[32,x]+F[56,x]
F[32,x] = F[10,x]*F[33,x]
F[33,x] = F[34,x]+F[39,x]
F[34,x] = F[35,x]+F[6,x]
F[35,x] = F[36,x]
F[36,x] = F[10,x]*F[37,x]
F[37,x] = F[10,x]+F[38,x]
F[38,x] = F[36,x]
F[39,x] = F[31,x]+F[40,x]
F[40,x] = 2*F[16,x]+F[41,x]+F[45,x]
F[41,x] = F[10,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[35,x]
F[44,x] = F[40,x]
F[45,x] = F[10,x]*F[46,x]
F[46,x] = F[47,x]+F[51,x]
F[47,x] = F[16,x]+F[48,x]+F[50,x]
F[48,x] = F[10,x]*F[49,x]
F[49,x] = F[37,x]+F[46,x]
F[50,x] = F[10,x]*F[21,x]
F[51,x] = 2*F[16,x]+F[45,x]+F[52,x]
F[52,x] = F[10,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[38,x]
F[55,x] = F[51,x]
F[56,x] = F[10,x]*F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[21,x]+F[47,x]
F[59,x] = F[25,x]+F[60,x]
F[60,x] = F[16,x]+F[52,x]+F[61,x]+F[63,x]
F[61,x] = F[10,x]*F[62,x]
F[62,x] = F[47,x]+F[60,x]
F[63,x] = F[10,x]*F[25,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{20}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{11}\! \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) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{15}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{19}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right)+F_{15}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{31}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{14}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right)+F_{26}\! \left(x \right)+F_{30}\! \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_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)
F_{30}\! \left(x \right) = F_{10}\! \left(x \right) F_{24}\! \left(x \right)
F_{31}\! \left(x \right) = F_{16}\! \left(x \right)+F_{32}\! \left(x \right)+F_{56}\! \left(x \right)
F_{32}\! \left(x \right) = F_{10}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{39}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{6}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{10}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{31}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{41}\! \left(x \right)+F_{45}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{35}\! \left(x \right)
F_{44}\! \left(x \right) = F_{40}\! \left(x \right)
F_{45}\! \left(x \right) = F_{10}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{51}\! \left(x \right)
F_{47}\! \left(x \right) = F_{16}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{37}\! \left(x \right)+F_{46}\! \left(x \right)
F_{50}\! \left(x \right) = F_{10}\! \left(x \right) F_{21}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{45}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{10}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{38}\! \left(x \right)
F_{55}\! \left(x \right) = F_{51}\! \left(x \right)
F_{56}\! \left(x \right) = F_{10}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{21}\! \left(x \right)+F_{47}\! \left(x \right)
F_{59}\! \left(x \right) = F_{25}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{16}\! \left(x \right)+F_{52}\! \left(x \right)+F_{61}\! \left(x \right)+F_{63}\! \left(x \right)
F_{61}\! \left(x \right) = F_{10}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{47}\! \left(x \right)+F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_20(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_10(x)*F_8(x))
Eq(F_8(x), F_11(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_15(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_14(x))
Eq(F_14(x), F_1(x) + F_12(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_19(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_10(x)*F_18(x))
Eq(F_18(x), F_10(x) + F_15(x))
Eq(F_19(x), F_10(x)*F_12(x))
Eq(F_20(x), F_21(x) + F_31(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_10(x)*F_23(x))
Eq(F_23(x), F_14(x) + F_24(x))
Eq(F_24(x), F_21(x) + F_25(x))
Eq(F_25(x), F_16(x) + F_26(x) + F_30(x))
Eq(F_26(x), F_10(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_12(x))
Eq(F_29(x), F_25(x))
Eq(F_30(x), F_10(x)*F_24(x))
Eq(F_31(x), F_16(x) + F_32(x) + F_56(x))
Eq(F_32(x), F_10(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_39(x))
Eq(F_34(x), F_35(x) + F_6(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_10(x)*F_37(x))
Eq(F_37(x), F_10(x) + F_38(x))
Eq(F_38(x), F_36(x))
Eq(F_39(x), F_31(x) + F_40(x))
Eq(F_40(x), 2*F_16(x) + F_41(x) + F_45(x))
Eq(F_41(x), F_10(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_35(x))
Eq(F_44(x), F_40(x))
Eq(F_45(x), F_10(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_51(x))
Eq(F_47(x), F_16(x) + F_48(x) + F_50(x))
Eq(F_48(x), F_10(x)*F_49(x))
Eq(F_49(x), F_37(x) + F_46(x))
Eq(F_50(x), F_10(x)*F_21(x))
Eq(F_51(x), 2*F_16(x) + F_45(x) + F_52(x))
Eq(F_52(x), F_10(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_38(x))
Eq(F_55(x), F_51(x))
Eq(F_56(x), F_10(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_21(x) + F_47(x))
Eq(F_59(x), F_25(x) + F_60(x))
Eq(F_60(x), F_16(x) + F_52(x) + F_61(x) + F_63(x))
Eq(F_61(x), F_10(x)*F_62(x))
Eq(F_62(x), F_47(x) + F_60(x))
Eq(F_63(x), F_10(x)*F_25(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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]]}], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 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