0123_0132_1230_2103
Counting sequence:
1, 1, 2, 6, 20, 61, 171, 470, 1271, 3402, 9044, 23930, 63114, 166089, 436398, 1145395, 3003999, 7874349, 20633297, 54051750, 141570157, 370747382, 970835062, 2542057742, 6655889836, 17426626449, 45625855804, 119454373611, 312743578653, 818787974913, 2143641704923, 5612176424882, 14692959826151, 38466835953862, 100707792477180, 263656991076142, 690264007691746, 1807136552979805, 4731148448767342, 12386313938763103, 32427802831463231, 84897111962528409, 222263565072405961, 581893642141816522, 1523417469663356437, 3988358966061976634, 10441659794933737190, 27336621092674435338, 71568204722649656796, 187367995355180987149, 490535785536295045124, 1284239368966572428767, 3362182335549598714293, 8802307663774670208245, 23044740703765903158235, 60331914535793153481502, 157951003065967609243239, 413521094960725331668018, 1082612282365448209352628, 2834315753145828829358450, 7420334978930103292702922, 19426689187061995419290913, 50859732588541671882659198, 133152508590124388530696859, 348597793203096165299471535, 912640871058275936177257461, 2389324820043669511933890897, 6255333589205046968725585150, 16376675947814835460855164237, 42874694254687075718254967214, 112247406817069686434038266662, 293867526198036258695015721366, 769355171779824275806708374940, 2014197989146559324732093298809, 5273238795669275915663410782940, 13805518397878598581694661684323, 36143316397998394962137920061149, 94624430796175213814146806186009, 247729975990635079281884074609195, 648565497175928359475232047231898, 1697966515537514794898547980476023, 4445334049437286989220456013288334, 11638035632775580267961326721460988, 30468772848891723669616810845648326, 79768282913903765655040943291201858, 208836075892827252159809649860300821, 546739944764592114457796761292315742, 1431383758400975068625444857327322983, 3747411330438380871328114419835288495, 9810850232914255426313747989637469505, 25685139368304547045889419968992358889, 67244567871999683010495228533859408322, 176048564247695048803968122256479833109, 460901124871086469157198001895909098066, 1206654810365566208540927320331990236342, 3159063306225615558913046116285027360306, 8270535108311286726274763486269129376236, 21652542018708256130308560394354106824661, 56687090947813502835572248363555940435012, 148408730824732291315803383872654247405559, 388539101526383442732551749149341830100373
Generating function in Maple syntax:
-(x^9-x^8-x^7-4*x^6+2*x^5+5*x^4+2*x^3+x^2-3*x+1)/(x^2-3*x+1)/(x^3+x^2+x-1)
Generating function in latex syntax:
-\frac{x^{9}-x^{8}-x^{7}-4 x^{6}+2 x^{5}+5 x^{4}+2 x^{3}+x^{2}-3 x +1}{\left(x^{2}-3 x +1\right) \left(x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(-x**9 + x**8 + x**7 + 4*x**6 - 2*x**5 - 5*x**4 - 2*x**3 - x**2 + 3*x - 1)/((x**2 - 3*x + 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x^3+x^2+x-1)*F(x)+x^9-x^8-x^7-4*x^6+2*x^5+5*x^4+2*x^3+x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+x^{9}-x^{8}-x^{7}-4 x^{6}+2 x^{5}+5 x^{4}+2 x^{3}+x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,n = 2,2,n = 3,6,n = 4,20,1/2640*(((70*I+10*11^(1/2))*3^(1/2)+30*I*11^(1/2)+70)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1120+((-545*I-95*11^(1/2))*3^(1/2)+285*I*11^(1/2)+545)*(17+3*11^(1/2)*3^(1/2))^(2/3))*(1/24*((17*I+3*11^(1/2))*3^(1/2)-9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/2640*(((545*I-95*11^(1/2))*3^(1/2)-285*I*11^(1/2)+545)*(17+3*11^(1/2)*3^(1/2))^(2/3)-1120+((-70*I+10*11^(1/2))*3^(1/2)-30*I*11^(1/2)+70)*(17+3*11^(1/2)*3^(1/2))^(1/3))*(1/24*((-17*I+3*11^(1/2))*3^(1/2)+9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)+1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/2640*((190*11^(1/2)*3^(1/2)-1090)*(17+3*11^(1/2)*3^(1/2))^(2/3)-20*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(1/2)-140*(17+3*11^(1/2)*3^(1/2))^(1/3)-1120)*(1/3*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3+17/12*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/4*(17+3*11^(1/2)*3^(1/2))^(2/3)*11^(1/2)*3^(1/2))^(-n)+1/2640*(-7104*5^(1/2)+17520)*(3/2-1/2*5^(1/2))^(-n)+1/2640*(7104*5^(1/2)+17520)*(3/2+1/2*5^(1/2))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ 6 & n =3 \\ 20 & n =4 \\ \frac{\left(\left(\left(70 \,\mathrm{I}+10 \sqrt{11}\right) \sqrt{3}+30 \,\mathrm{I} \sqrt{11}+70\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-1120+\left(\left(-545 \,\mathrm{I}-95 \sqrt{11}\right) \sqrt{3}+285 \,\mathrm{I} \sqrt{11}+545\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{2640}\\+\\\frac{\left(\left(\left(545 \,\mathrm{I}-95 \sqrt{11}\right) \sqrt{3}-285 \,\mathrm{I} \sqrt{11}+545\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}-1120+\left(\left(-70 \,\mathrm{I}+10 \sqrt{11}\right) \sqrt{3}-30 \,\mathrm{I} \sqrt{11}+70\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{2640}\\+\\\frac{\left(\left(190 \sqrt{11}\, \sqrt{3}-1090\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}-20 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-140 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-1120\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{2640}\\+\frac{\left(-7104 \sqrt{5}+17520\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{2640}+\frac{\left(7104 \sqrt{5}+17520\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{2640} & \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) = 61
a(6) = 171
a(7) = 470
a(8) = 1271
a(9) = 3402
a(n+5) = a(n)-2*a(n+1)-a(n+2)-3*a(n+3)+4*a(n+4), n >= 10
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) = 61
a \! \left(6\right) = 171
a \! \left(7\right) = 470
a \! \left(8\right) = 1271
a \! \left(9\right) = 3402
a \! \left(n +5\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)-3 a \! \left(n +3\right)+4 a \! \left(n +4\right), \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14848/
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[19,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[18,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[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[43,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[32,x]
F[23,x] = F[24,x]+F[7,x]
F[24,x] = F[13,x]+F[14,x]+F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[31,x]
F[27,x] = F[28,x]+F[7,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[16,x]
F[31,x] = F[24,x]
F[32,x] = F[33,x]+F[38,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[2,x]
F[37,x] = F[33,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[20,x]
F[42,x] = F[38,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[50,x]
F[45,x] = F[2,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[28,x]+F[4,x]
F[50,x] = F[20,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[12,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_{19}\! \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_{18}\! \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_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{43}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{32}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{16}\! \left(x \right)
F_{31}\! \left(x \right) = F_{24}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{38}\! \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_{2}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{20}\! \left(x \right)
F_{42}\! \left(x \right) = F_{38}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{50}\! \left(x \right)
F_{45}\! \left(x \right) = F_{2}\! \left(x \right)+F_{46}\! \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_{49}\! \left(x \right) = F_{28}\! \left(x \right)+F_{4}\! \left(x \right)
F_{50}\! \left(x \right) = F_{20}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \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_19(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_18(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_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_43(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_32(x))
Eq(F_23(x), F_24(x) + F_7(x))
Eq(F_24(x), F_13(x) + F_14(x) + F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_31(x))
Eq(F_27(x), F_28(x) + F_7(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_16(x))
Eq(F_31(x), F_24(x))
Eq(F_32(x), F_33(x) + F_38(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_2(x))
Eq(F_37(x), F_33(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_20(x))
Eq(F_42(x), F_38(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_50(x))
Eq(F_45(x), F_2(x) + F_46(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))
Eq(F_49(x), F_28(x) + F_4(x))
Eq(F_50(x), F_20(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_12(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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